%!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: arXiv:1502.01875v1 [math.FA] 6 Feb 2015 %%CreationDate: Tue Mar 16 14:25:11 2021 %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: EUFM10 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z ExtensionPaper_Feb04.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2021.03.16:1425 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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end 3633 3909 a Fu(.)340 4055 y(The)k(other)g(example)f(that)g(w)m(e)i(consider)f(is)f (that)g(of)h(balls)e(in)h(a)g(Hilb)s(ert)f(space,)k(endo)m(w)m(ed)240 4200 y(with)i(their)h(w)m(eak)h(top)s(ology)-8 b(.)41 b(Corson)33 b(and)f(Lindenstrauss)h([)2534 4200 y SDict begin H.S end 2534 4200 a Black Fu(CL)p Black 2665 4132 a SDict begin H.R end 2665 4132 a 2665 4200 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.CL) cvn H.B /ANN pdfmark end 2665 4200 a Fu(,)f(Prop)s(osition)e(1])i(sho)m(w)m(ed)240 4345 y(that,)f(for)f(the)h(unit)f(ball)e Fs(B)1255 4360 y Fn(H)1353 4345 y Fu(of)i(a)g(nonseparable)h(Hilb)s(ert)e(space)i Fs(H)38 b Fu(and)31 b(an)m(y)g(0)d Fs(<)f(\025)h(<)f(\026)p Fu(,)240 4491 y(there)39 b(exists)f(no)g(w)m(eak-con)m(tin)m(uous)i (retraction)d Fs(r)i Fu(:)e Fs(\026B)2383 4506 y Fn(H)2487 4491 y Fr(!)g Fs(\025B)2755 4506 y Fn(H)2822 4491 y Fu(,)i(i.e.,)g(a)f (map)f Fs(r)j Fu(suc)m(h)240 4636 y(that)32 b Fs(r)s 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Fu(\()p Fs(\016)311 4872 y Fn(x)355 4857 y Fu(\))k(with)f(the)h(k)m (ey)h(prop)s(ert)m(y)f(that)f Fs(')1665 4872 y Fn(T)1720 4857 y Fu(\()p Fs(y)t Fu(\))27 b(=)g Fs(\016)2021 4872 y Fn(y)2095 4857 y Fu(when)34 b Fs(y)d Fr(2)d Fs(L)p Fu(.)44 b(The)33 b(function)f Fs(')3306 4872 y Fn(T)3394 4857 y Fu(can)120 5002 y(b)s(e)k(view)m(ed)g(as)g Fq(gener)-5 b(alize)g(d)36 b(r)-5 b(etr)g(action)42 b Fu(that)35 b(sends)i(eac)m(h)g(p)s(oin)m(t)d(of)h Fs(K)42 b Fu(to)35 b(a)g(measure)h(on)120 5147 y Fs(L)d Fu(\(instead)f(of)g(a)h(p)s(oin)m (t)e(of)i Fs(L)p Fu(\).)p Black Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 232 -8 a SDict begin H.S end 232 -8 a 232 -8 a SDict begin H.R end 232 -8 a 232 -8 a SDict begin [/View 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SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ER) cvn H.B /ANN pdfmark end 2797 1021 a Fu(,)44 b(Lemma)c(1.1],)j([)3474 1021 y SDict begin H.S end 3474 1021 a Black Fu(BM)p Black 3633 953 a SDict begin H.R end 3633 953 a 3633 1021 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BM) cvn H.B /ANN pdfmark end 3633 1021 a Fu(,)240 1166 y(Lemma)31 b(3.2])240 1375 y SDict begin H.S end 240 1375 a 240 1375 a SDict begin 17.5 H.A end 240 1375 a 240 1375 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.1) cvn /DEST pdfmark end 240 1375 a Black FA(Lemma)41 b(2.1.)p Black 43 w Fq(L)-5 b(et)37 b Fs(n)32 b Fr(2)f Fs(!)t Fq(,)37 b Fs(X)44 b Fq(b)-5 b(e)37 b(a)f(set)h(of)g(c)-5 b(ar)g(dinality)36 b Fr(\025)c(@)2600 1390 y Fn(n)2646 1375 y Fq(,)37 b(and)g Fs(S)g Fu(:)31 b([)p Fs(X)8 b Fu(])3203 1339 y Fn(n)3281 1375 y Fr(!)31 b Fu([)p Fs(X)8 b Fu(])3555 1339 y Fn(<!)240 1520 y Fq(b)-5 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Fs(X)g Fu(])383 2337 y Fn(<!)522 2373 y Fq(b)-5 b(e)33 b(a)g(map)g(such)g(that)h Fs(S)6 b Fu(\()p Fs(A)p Fu(\))18 b Fr(\\)i Fs(A)28 b Fu(=)f Fr(;)33 b Fq(for)h(every)f Fs(A)p Fq(.)44 b(Then)33 b(ther)-5 b(e)33 b(exists)g Fs(Y)49 b Fr(2)28 b Fu([)p Fs(X)8 b Fu(])3620 2337 y Fn(p)240 2518 y Fq(such)35 b(that)g Fs(S)6 b Fu(\()p Fs(A)p Fu(\))22 b Fr(\\)g Fs(Y)49 b Fu(=)28 b Fr(;)35 b Fq(for)f(al)5 b(l)35 b Fs(A)28 b Fr(2)g Fu([)p Fs(Y)21 b Fu(])1903 2482 y Fk(\024)p Fn(m)2025 2518 y Fq(.)p Black 240 2727 a(Pr)-5 b(o)g(of.)p Black 41 w Fu(Apply)41 b(Lemma)1196 2727 y SDict begin H.S end 1196 2727 a Black Fu(2.1)p Black 1321 2663 a SDict begin H.R end 1321 2663 a 1321 2727 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.1) cvn H.B /ANN pdfmark end 1321 2727 a 41 w Fu(to)g Fs(n)h Fu(=)h Fs(p)28 b Fr(\000)g Fu(1,)43 b(and)e(the)h(function)f Fs(S)2842 2691 y Fk(0)2907 2727 y Fu(:)i([)p Fs(X)8 b Fu(])3120 2691 y Fn(n)3209 2727 y Fr(\000)-16 b(!)42 b Fu([)p Fs(X)8 b Fu(])3555 2691 y Fn(<!)240 2872 y Fu(giv)m(en)33 b(b)m(y)g Fs(S)696 2836 y Fk(0)719 2872 y Fu(\()p Fs(A)p Fu(\))28 b(=)1000 2797 y Fm(S)1083 2901 y Fn(B)s Fk(2)p Fl([)p Fn(A)p Fl(])1279 2882 y Fi(\024)p Fh(m)1405 2872 y Fs(S)6 b Fu(\()p Fs(B)f Fu(\).)1930 b Fg(\003)340 3067 y Fu(W)-8 b(e)31 b(shall)e(also)h(need)i (a)f(v)m(ersion)g(of)g(this)f(lemma)f(that)h(holds)h(for)f(an)m(y)i (uncoun)m(table)f(set.)240 3276 y SDict begin H.S end 240 3276 a 240 3276 a SDict begin 17.5 H.A end 240 3276 a 240 3276 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.3) cvn /DEST pdfmark end 240 3276 a Black FA(Lemma)47 b(2.3.)p Black 46 w Fq(L)-5 b(et)42 b Fs(X)49 b Fq(b)-5 b(e)41 b(an)g(unc)-5 b(ountable)41 b(set,)i Fs(n)d(<)g(!)t Fq(,)i(and)f Fs(S)k Fu(:)40 b([)p Fs(X)8 b Fu(])3036 3239 y Fn(<!)3181 3276 y Fr(\000)-16 b(!)40 b Fu([)p Fs(X)8 b Fu(])3525 3239 y Fn(<!)3630 3276 y Fq(.)240 3421 y(Then)28 b(ther)-5 b(e)29 b(exists)f Fs(Z)35 b Fu(=)27 b Fr(f)p Fs(z)1285 3436 y Fl(1)1325 3421 y Fs(;)17 b(:)g(:)g(:)f(;)h(z)1589 3436 y Fn(n)1636 3421 y Fr(g)27 b(2)h Fu([)p Fs(X)8 b Fu(])1950 3385 y Fn(n)2026 3421 y Fq(such)29 b(that)g Fs(z)2480 3436 y Fn(j)2544 3421 y Fr(62)f Fs(S)6 b Fu(\()p Fr(f)p Fs(z)2837 3436 y Fn(i)2893 3421 y Fu(:)28 b Fs(i)g Fr(2)g Fs(I)8 b Fr(g)p Fu(\))28 b Fq(whenever)240 3566 y Fs(j)34 b(<)27 b Fu(min)o(\()p Fs(I)8 b Fu(\))34 b Fq(or)h Fs(j)f(>)27 b Fu(max\()p Fs(I)8 b Fu(\))p Fq(.)p Black 240 3775 a(Pr)-5 b(o)g(of.)p Black 41 w Fu(W)d(e)43 b(start)g(the)g(construction)g(with)f(a)g(coun)m (table)h(in\014nite)f(set)h Fs(Y)3042 3790 y Fl(1)3126 3775 y Fr(\032)i Fs(X)8 b Fu(.)74 b(Then)240 3920 y(inductiv)m(ely)-8 b(,)32 b(w)m(e)i(c)m(ho)s(ose)f(a)g(coun)m(table)f(in\014nite)g(set)h Fs(Y)2276 3935 y Fn(j)2340 3920 y Fr(\032)28 b Fs(X)40 b Fu(for)32 b Fs(j)i Fu(=)28 b(2)p Fs(;)17 b(:)g(:)g(:)e(;)i(n)33 b Fu(suc)m(h)h(that)1042 4192 y(\()p Fs(?)p Fu(\))65 b Fs(Y)1289 4207 y Fn(j)1347 4192 y Fr(\\)1436 4098 y Fm([)1563 4022 y(\()1643 4192 y Fs(S)6 b Fu(\()p Fs(A)p Fu(\))28 b(:)f Fs(A)h Fr(2)2135 4022 y Fm(")2194 4098 y([)2193 4308 y Fn(i<j)2321 4192 y Fs(Y)2378 4207 y Fn(i)2406 4022 y Fm(#)2464 4044 y Fn(<!)2569 4022 y Fm(\))2677 4192 y Fu(=)g Fr(;)p Fs(:)240 4465 y Fu(Then,)47 b(b)m(y)d(rev)m(erse)h (induction)d(\(starting)g(for)h Fs(j)51 b Fu(=)46 b Fs(n)d Fu(and)g(then)h Fs(n)29 b Fr(\000)h Fu(1)p Fs(;)17 b(n)29 b Fr(\000)h Fu(2)p Fs(;)17 b(:)g(:)g(:)o Fu(\))43 b(w)m(e)240 4610 y(c)m(ho)s(ose)33 b Fs(z)594 4625 y Fn(j)659 4610 y Fr(2)28 b Fs(Y)810 4625 y Fn(j)879 4610 y Fu(suc)m(h)34 b(that)968 4806 y(\()p Fs(??)p Fu(\))65 b Fs(z)1252 4821 y Fn(j)1316 4806 y Fr(62)1410 4711 y Fm([)1538 4725 y(\010)1596 4806 y Fs(S)6 b Fu(\()p Fs(A)p Fu(\))27 b(:)h Fs(A)g Fr(2)g Fu([)p Fr(f)p Fs(z)2210 4821 y Fn(j)t Fl(+1)2337 4806 y Fs(;)17 b(:)g(:)g(:)f(;)h(z)2601 4821 y Fn(n)2648 4806 y Fr(g)p Fu(])2725 4758 y Fn(<!)2830 4725 y Fm(\011)2905 4806 y Fs(:)240 5002 y Fu(Then,)44 b(\()p Fs(?)p Fu(\))c(guaran)m(tees) h(the)g(statemen)m(t)g(of)f(the)h(lemma)d(when)k Fs(j)47 b(>)41 b Fu(max\()p Fs(I)8 b Fu(\),)42 b(and)f(\()p Fs(??)p Fu(\))240 5147 y(when)34 b Fs(j)f(<)28 b Fu(min)n(\()p Fs(I)8 b Fu(\).)p Black Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 112 -8 a SDict begin H.S end 112 -8 a 112 -8 a SDict begin H.R end 112 -8 a 112 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end 112 -8 a Black 442 66 a Fo(EXTENSION)32 b(OPERA)-6 b(TORS)32 b(ON)g(BALLS)h(AND)f(ON)h (SP)-6 b(A)n(CES)33 b(OF)g(FINITE)g(SETS)284 b(5)p Black 3463 266 a Fg(\003)120 657 y SDict begin H.S end 120 657 a 120 657 a SDict begin 17.5 H.A end 120 657 a 120 657 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 120 657 a 351 803 a Fu(3.)49 b Ft(A)38 b(canonical)g(extension)f(opera)-7 b(tor)38 b(between)e(sp)-7 b(a)n(ces)36 b Fs(\033)3049 818 y Fn(m)3117 803 y Fu(\(2)3204 767 y Fn(X)3271 803 y Fu(\))120 1006 y SDict begin H.S end 120 1006 a 120 1006 a SDict begin 17.5 H.A end 120 1006 a 120 1006 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.1) cvn /DEST pdfmark end 120 1006 a Black FA(Theorem)k(3.1.)p Black 43 w Fq(Consider)35 b Fs(m)c(<)f(n)g(<)h(!)39 b Fq(and)d(a)g(set)h Fs(X)8 b Fq(.)49 b(Then,)35 b(we)h(have)g(an)g (extension)120 1151 y(op)-5 b(er)g(ator)35 b Fs(T)561 1166 y Fn(X)656 1151 y Fu(:)28 b Fs(C)7 b Fu(\()p Fs(\033)881 1166 y Fn(m)947 1151 y Fu(\(2)1034 1115 y Fn(X)1102 1151 y Fu(\)\))27 b Fr(\000)-16 b(!)27 b Fs(C)7 b Fu(\()p Fs(\033)1563 1166 y Fn(n)1611 1151 y Fu(\(2)1698 1115 y Fn(X)1765 1151 y Fu(\)\))34 b Fq(given)g(by)698 1516 y Fs(T)755 1531 y Fn(X)822 1516 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)1056 1531 y Fn(A)1113 1516 y Fu(\))28 b(=)1361 1421 y Fm(X)1282 1641 y Fn(B)s Fk(2)p Fl([)p Fn(A)p Fl(])1478 1622 y Fi(\024)p Fh(m)1584 1516 y Fu(\()p Fr(\000)p Fu(1\))1786 1475 y Fn(m)p Fk(\000j)p Fn(B)s Fk(j)2004 1375 y Fm(\022)2077 1448 y Fr(j)p Fs(A)p Fr(j)22 b(\000)g(j)p Fs(B)5 b Fr(j)22 b(\000)h Fu(1)2184 1584 y Fs(m)f Fr(\000)h(j)p Fs(B)5 b Fr(j)2632 1375 y Fm(\023)2706 1516 y Fs(f)11 b Fu(\()p Fs(\037)2864 1531 y Fn(B)2924 1516 y Fu(\))120 1937 y Fq(if)35 b Fr(j)p Fs(A)p Fr(j)27 b Fs(>)g(m)p Fq(,)35 b(and)g(by)g Fs(T)998 1952 y Fn(X)1065 1937 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)1299 1952 y Fn(A)1356 1937 y Fu(\))28 b(=)f Fs(f)11 b Fu(\()p Fs(\037)1683 1952 y Fn(A)1740 1937 y Fu(\))35 b Fq(if)f Fr(j)p Fs(A)p Fr(j)27 b(\024)i Fs(m)p Fq(.)220 2378 y Fu(The)35 b(pro)s(of)f(of)g(Theorem) 1206 2378 y SDict begin H.S end 1206 2378 a Black Fu(3.1)p Black 1331 2314 a SDict begin H.R end 1331 2314 a 1331 2378 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.1) cvn H.B /ANN pdfmark end 1331 2378 a 34 w Fu(will)e(require)j(a)g(n)m (um)m(b)s(er)g(of)f(tec)m(hnical)g(lemmas)f(sho)m(wing)120 2523 y(some)f(com)m(binatorial)d(iden)m(tities.)42 b(F)-8 b(or)32 b(an)m(y)h(natural)e(n)m(um)m(b)s(ers)i Fs(p;)17 b(q)t(;)g(r)m(;)g(s;)g(t)31 b Fu(suc)m(h)j(that)e Fs(p)c Fr(\024)120 2669 y Fu(min)o(\()p Fs(q)t(;)17 b(r)m(;)g(s;)g(t)p Fu(\))p Fs(;)g(t)26 b Fr(\024)j Fs(s)j Fu(w)m(e)i(de\014ne)893 3176 y(\002\()p Fs(p;)17 b(q)t(;)g(r)m(;)g(s;)g(t)p Fu(\))27 b(=)1624 3047 y Fn(p)1569 3081 y Fm(X)1584 3291 y Fn(i)p Fl(=0)1713 3176 y Fu(\()p Fr(\000)p Fu(1\))1915 3135 y Fn(q)r Fk(\000)p Fn(i)2032 3036 y Fm(\022)2106 3109 y Fs(r)2112 3244 y(i)2152 3036 y Fm(\023)q(\022)2299 3109 y Fs(s)22 b Fr(\000)h Fs(i)2304 3244 y(t)g Fr(\000)f Fs(i)2500 3036 y Fm(\023)2573 3176 y Fs(:)220 3566 y Fu(Observ)m(e)34 b(that)1013 3897 y SDict begin H.S end 1013 3897 a 1013 3897 a SDict begin 17.5 H.A end 1013 3897 a 1013 3897 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.1) cvn /DEST pdfmark end 1013 3897 a Fu(\002\()p Fs(p;)17 b(q)26 b Fu(+)c(1)p Fs(;)17 b(r)m(;)g(s;)g(t)p Fu(\))27 b(=)g Fr(\000)p Fu(\002\()p Fs(p;)17 b(q)t(;)g(r)m(;)g(s;)g(t) p Fu(\))p Black -2361 w(\(3.1\))p Black 120 4193 a SDict begin H.S end 120 4193 a 120 4193 a SDict begin 17.5 H.A end 120 4193 a 120 4193 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.2) cvn /DEST pdfmark end 120 4193 a Black FA(Lemma)34 b(3.2.)p Black 38 w Fq(F)-7 b(or)30 b(any)h(natur)-5 b(al)31 b(numb)-5 b(ers)30 b Fs(p;)17 b(q)t(;)g(r)m(;)g(s;)g(t)30 b Fq(such)g(that)h Fs(p)d Fr(\024)g Fu(min)o(\()p Fs(q)t(;)17 b(r)m(;)g(s;)g(t)p Fu(\))p Fs(;)g(t)27 b Fr(\024)120 4338 y Fs(s)35 b Fq(we)f(have)g(the)h(fol)5 b(lowing)34 b(identities)771 4670 y SDict begin H.S end 771 4670 a 771 4670 a SDict begin 17.5 H.A end 771 4670 a 771 4670 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 771 4670 a Fu(\002\()p Fs(p;)17 b(q)t(;)g(r)24 b Fu(+)e(1)p Fs(;)17 b(s)22 b Fu(+)g(1)p Fs(;)17 b(p)p Fu(\))82 b(=)h(\002\()p Fs(p;)17 b(q)t(;)g(r)m(;)g(s;)g(p)p Fu(\))p Fs(;)p Black -2382 w Fu(\(3.2\))p Black 445 4852 a(\002\()p Fs(p)22 b Fu(+)g(1)p Fs(;)17 b(q)25 b Fu(+)d(1)p Fs(;)17 b(p)22 b Fu(+)g(1)p Fs(;)17 b(s)22 b Fu(+)g(1)p Fs(;)17 b(t)p Fu(\))82 b(=)h Fr(\000)p Fu(\002\()p Fs(p;)17 b(q)t(;)g(p;)g(s;)g(t)p Fu(\))99 b Fq(for)34 b Fs(p)28 b Fr(\024)g Fs(t)22 b Fr(\000)h Fu(1)p Fs(:)p Black -3095 w Fu(\(3.3\))p Black Black 120 5147 a Fq(Pr)-5 b(o)g(of.)p Black 41 w Fu(W)d(e)33 b(start)g(with)f(the)h(\014rst)g(iden)m(tit)m(y) p Black Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 232 -8 a SDict begin H.S end 232 -8 a 232 -8 a SDict begin H.R end 232 -8 a 232 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end 232 -8 a Black 240 70 a Fo(6)824 b(ANTONIO)24 b(A)-9 b(VIL)1693 52 y(\023)1686 70 y(ES,)25 b(WITOLD)g(MAR)n(CISZEWSKI)p Black 998 1408 a Fu(\002\()p Fs(p;)17 b(q)t(;)g(r)24 b Fu(+)e(1)p Fs(;)17 b(s)k Fu(+)h(1)p Fs(;)17 b(p)p Fu(\))27 b(=)2085 1279 y Fn(p)2031 1313 y Fm(X)2046 1523 y Fn(i)p Fl(=0)2175 1408 y Fu(\()p Fr(\000)p Fu(1\))2377 1367 y Fn(q)r Fk(\000)p Fn(i)2494 1267 y Fm(\022)2567 1340 y Fs(r)e Fu(+)d(1)2658 1476 y Fs(i)2783 1267 y Fm(\023\022)2929 1340 y Fs(s)g Fu(+)g(1)g Fr(\000)h Fs(i)3012 1476 y(p)f Fr(\000)h Fs(i)3299 1267 y Fm(\023)361 1745 y Fu(=)28 b(\()p Fr(\000)p Fu(1\))667 1704 y Fn(q)705 1604 y Fm(\022)778 1677 y Fs(r)d Fu(+)d(1)862 1813 y(0)994 1604 y Fm(\023\022)1141 1677 y Fs(s)g Fu(+)g(1)1224 1813 y Fs(p)1356 1604 y Fm(\023)1451 1745 y Fu(+)1603 1616 y Fn(p)1549 1650 y Fm(X)1564 1860 y Fn(i)p Fl(=1)1693 1745 y Fu(\()p Fr(\000)p Fu(1\))1895 1704 y Fn(q)r Fk(\000)p Fn(i)2029 1604 y Fm(\022\022)2254 1677 y Fs(r)2175 1813 y(i)h Fr(\000)f Fu(1)2379 1604 y Fm(\023)2475 1745 y Fu(+)2573 1604 y Fm(\022)2646 1677 y Fs(r)2653 1813 y(i)2693 1604 y Fm(\023\023)17 b(\022)2929 1677 y Fs(s)22 b Fu(+)g(1)g Fr(\000)h Fs(i)3012 1813 y(p)f Fr(\000)h Fs(i)3299 1604 y Fm(\023)1489 2059 y Fu(=)28 b(\()p Fr(\000)p Fu(1\))1795 2018 y Fn(q)1833 1919 y Fm(\022)1906 1992 y Fs(s)22 b Fu(+)g(1)1989 2127 y Fs(p)2121 1919 y Fm(\023)2217 2059 y Fu(+)g(\()p Fr(\000)p Fu(1\))2517 2018 y Fn(q)r Fk(\000)p Fl(1)2645 1919 y Fm(\022)2719 1992 y Fs(r)2718 2127 y Fu(0)2767 1919 y Fm(\023\022)2914 1992 y Fs(s)g Fu(+)g(1)g Fr(\000)g Fu(1)2997 2127 y Fs(p)g Fr(\000)g Fu(1)3299 1919 y Fm(\023)1118 2380 y Fu(+)1220 2251 y Fn(p)p Fk(\000)p Fl(1)1211 2285 y Fm(X)1226 2495 y Fn(i)p Fl(=1)1355 2380 y Fu(\()p Fr(\000)p Fu(1\))1557 2339 y Fn(q)r Fk(\000)p Fn(i)1674 2239 y Fm(\022)1747 2312 y Fs(r)1754 2448 y(i)1794 2239 y Fm(\023)17 b(\022\022)2030 2312 y Fs(s)23 b Fu(+)f(1)f Fr(\000)i Fs(i)2113 2448 y(p)g Fr(\000)f Fs(i)2400 2239 y Fm(\023)2496 2380 y Fr(\000)2595 2239 y Fm(\022)2669 2312 y Fs(s)g 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3288 y Fn(q)r Fk(\000)p Fn(p)2811 3189 y Fm(\022)2885 3262 y Fs(r)2884 3397 y(p)2933 3189 y Fm(\023\022)3081 3262 y Fs(s)f Fr(\000)h Fs(p)3080 3397 y(p)f Fr(\000)g Fs(p)3299 3189 y Fm(\023)1511 3635 y Fu(=)27 b(\()p Fr(\000)p Fu(1\))1816 3594 y Fn(q)1854 3495 y Fm(\022)1929 3568 y Fs(r)1928 3704 y Fu(0)1976 3495 y Fm(\023)q(\022)2125 3568 y Fs(s)2123 3704 y(p)2172 3495 y Fm(\023)2268 3635 y Fu(+)2420 3507 y Fn(p)2366 3541 y Fm(X)2380 3751 y Fn(i)p Fl(=1)2509 3635 y Fu(\()p Fr(\000)p Fu(1\))2711 3594 y Fn(q)r Fk(\000)p Fn(i)2829 3495 y Fm(\022)2902 3568 y Fs(r)2909 3704 y(i)2949 3495 y Fm(\023\022)3097 3568 y Fs(s)22 b Fr(\000)g Fs(i)3095 3704 y(p)g Fr(\000)h Fs(i)3299 3495 y Fm(\023)1572 3973 y Fu(=)1730 3844 y Fn(p)1676 3878 y Fm(X)1691 4088 y Fn(i)p Fl(=0)1820 3973 y Fu(\()p Fr(\000)p Fu(1\))2022 3931 y Fn(q)r Fk(\000)p Fn(i)2139 3832 y Fm(\022)2212 3905 y Fs(r)2219 4041 y(i)2259 3832 y Fm(\023\022)2407 3905 y Fs(s)f Fr(\000)h Fs(i)2405 4041 y(p)g Fr(\000)f Fs(i)2609 3832 y Fm(\023)2711 3973 y Fu(=)27 b(\002\()p Fs(p;)17 b(q)t(;)g(r)m(;)g(s;)g(p)p Fu(\))340 5147 y(The)33 b(calculations)e(for)h(the)h(iden)m(tit)m(y)f (\()1787 5147 y SDict begin H.S end 1787 5147 a Black Fu(3.3)p Black 1912 5083 a SDict begin H.R end 1912 5083 a 1912 5147 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 1912 5147 a Fu(\))g(are)h(v)m(ery)h (similar)p Black Black eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 112 -8 a SDict begin H.S end 112 -8 a 112 -8 a SDict begin H.R end 112 -8 a 112 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end 112 -8 a Black 442 66 a Fo(EXTENSION)32 b(OPERA)-6 b(TORS)32 b(ON)g(BALLS)h(AND)f(ON)h (SP)-6 b(A)n(CES)33 b(OF)g(FINITE)g(SETS)284 b(7)p Black 350 544 a Fu(\002\()p Fs(p)22 b Fu(+)g(1)p Fs(;)17 b(q)25 b Fu(+)d(1)p Fs(;)17 b(p)22 b Fu(+)g(1)p Fs(;)17 b(s)22 b Fu(+)g(1)p Fs(;)17 b(t)p Fu(\))27 b(=)1719 416 y Fn(p)p Fl(+1)1710 450 y Fm(X)1724 660 y Fn(i)p Fl(=0)1853 544 y Fu(\()p Fr(\000)p Fu(1\))2055 503 y Fn(q)r Fl(+1)p Fk(\000)p Fn(i)2263 404 y Fm(\022)2336 477 y Fs(p)22 b Fu(+)g(1)2428 613 y Fs(i)2554 404 y Fm(\023\022)2701 477 y Fs(s)g Fu(+)g(1)g Fr(\000)g Fs(i)2790 613 y(t)h Fr(\000)f Fs(i)3070 404 y Fm(\023)414 881 y Fu(=)27 b Fr(\000)p Fu(\()p Fr(\000)p Fu(1\))796 840 y Fn(q)835 741 y Fm(\022)908 814 y Fs(s)22 b Fu(+)g(1)998 950 y Fs(t)1123 741 y Fm(\023)1218 881 y Fr(\000)1372 753 y Fn(p)1318 787 y Fm(X)1333 997 y Fn(i)p Fl(=1)1462 881 y Fu(\()p Fr(\000)p Fu(1\))1664 840 y Fn(q)r Fk(\000)p Fn(i)1798 741 y Fm(\022\022)2022 814 y Fs(p)1944 950 y(i)h Fr(\000)f Fu(1)2148 741 y Fm(\023)2244 881 y Fu(+)2342 741 y Fm(\022)2415 814 y Fs(p)2423 950 y(i)2464 741 y Fm(\023\023)17 b(\022)2701 814 y Fs(s)22 b Fu(+)g(1)g Fr(\000)g Fs(i)2790 950 y(t)h Fr(\000)f Fs(i)3070 741 y Fm(\023)599 1196 y Fu(+\()p Fr(\000)p Fu(1\))877 1155 y Fn(q)r Fk(\000)p Fn(p)1006 1055 y Fm(\022)1159 1128 y Fs(s)g Fr(\000)h Fs(p)1079 1264 y(t)f Fr(\000)h Fs(p)f Fr(\000)h Fu(1)1456 1055 y Fm(\023)1557 1196 y Fu(=)k Fr(\000)p Fu(\()p Fr(\000)p Fu(1\))1939 1155 y Fn(q)1978 1055 y Fm(\022)2051 1128 y Fs(s)22 b Fu(+)g(1)2141 1264 y Fs(t)2266 1055 y Fm(\023)2361 1196 y Fr(\000)h Fu(\()p Fr(\000)p Fu(1\))2663 1155 y Fn(q)r Fk(\000)p Fl(1)2791 1055 y Fm(\022)2945 1128 y Fs(s)2865 1264 y(t)f Fr(\000)h Fu(1)3070 1055 y Fm(\023)1065 1516 y Fr(\000)1169 1388 y Fn(p)p Fk(\000)p Fl(1)1159 1422 y Fm(X)1174 1632 y Fn(i)p Fl(=1)1303 1516 y Fu(\()p Fr(\000)p Fu(1\))1505 1475 y Fn(q)r Fk(\000)p Fn(i)1622 1376 y Fm(\022)1696 1449 y Fs(p)1704 1585 y(i)1745 1376 y Fm(\023)17 b(\022\022)1981 1449 y Fs(s)22 b Fu(+)g(1)g Fr(\000)h Fs(i)2071 1585 y(t)g Fr(\000)f Fs(i)2351 1376 y Fm(\023)2447 1516 y Fr(\000)2546 1376 y Fm(\022)2700 1449 y Fs(s)g Fr(\000)g Fs(i)2620 1585 y(t)g Fr(\000)h Fs(i)f Fr(\000)h Fu(1)2981 1376 y Fm(\023\023)1230 1831 y Fr(\000)p Fu(\()p Fr(\000)p Fu(1\))1509 1790 y Fn(q)r Fk(\000)p Fn(p)1638 1690 y Fm(\022)1711 1763 y Fs(s)f Fu(+)g(1)g Fr(\000)h Fs(p)1801 1899 y(t)f Fr(\000)h Fs(p)2097 1690 y Fm(\023)2192 1831 y Fu(+)f(\()p Fr(\000)p Fu(1\))2492 1790 y Fn(q)r Fk(\000)p Fn(p)2621 1690 y Fm(\022)2774 1763 y Fs(s)g Fr(\000)h Fs(p)2694 1899 y(t)f Fr(\000)h Fs(p)f Fr(\000)h Fu(1)3070 1690 y Fm(\023)586 2151 y Fu(=)28 b Fr(\000)p Fu(\()p Fr(\000)p Fu(1\))969 2110 y Fn(q)1007 2011 y Fm(\022)1081 2084 y Fs(s)1086 2220 y(t)1127 2011 y Fm(\023)1222 2151 y Fr(\000)1331 2023 y Fn(p)p Fk(\000)p Fl(1)1322 2057 y Fm(X)1337 2267 y Fn(i)p Fl(=1)1466 2151 y Fu(\()p Fr(\000)p Fu(1\))1668 2110 y Fn(q)r Fk(\000)p Fn(i)1785 2011 y Fm(\022)1858 2084 y Fs(p)1866 2220 y(i)1907 2011 y Fm(\023\022)2054 2084 y Fs(s)22 b Fr(\000)h Fs(i)2059 2220 y(t)g Fr(\000)f Fs(i)2255 2011 y Fm(\023)2350 2151 y Fr(\000)h Fu(\()p Fr(\000)p Fu(1\))2652 2110 y Fn(q)r Fk(\000)p Fn(p)2780 2011 y Fm(\022)2854 2084 y Fs(s)f Fr(\000)h Fs(p)2859 2220 y(t)g Fr(\000)f Fs(p)3070 2011 y Fm(\023)1179 2488 y Fu(=)27 b Fr(\000)1431 2360 y Fn(p)1376 2394 y Fm(X)1391 2604 y Fn(i)p Fl(=0)1520 2488 y Fu(\()p Fr(\000)p Fu(1\))1722 2447 y Fn(q)r Fk(\000)p Fn(i)1839 2348 y Fm(\022)1913 2421 y Fs(p)1921 2557 y(i)1962 2348 y Fm(\023\022)2108 2421 y Fs(s)22 b Fr(\000)h Fs(i)2114 2557 y(t)f Fr(\000)h Fs(i)2309 2348 y Fm(\023)2410 2488 y Fu(=)28 b Fr(\000)p Fu(\002\()p Fs(p;)17 b(q)t(;)g(p;)g(s;)g(t)p Fu(\))3463 2778 y Fg(\003)220 3011 y Fu(F)-8 b(or)31 b(an)m(y)j(natural)d(n)m(um)m (b)s(ers)j Fs(k)s(;)17 b(l)r(;)g(m)32 b Fu(suc)m(h)i(that)e Fs(l)e Fr(\025)f Fu(1)p Fs(;)17 b(m)27 b Fr(\024)h Fs(k)d Fu(+)d Fs(l)35 b Fu(w)m(e)f(de\014ne)689 3290 y(\010\()p Fs(k)s(;)17 b(l)r(;)g(m)p Fu(\))28 b(=)1266 3166 y Fn(m)1225 3196 y Fm(X)1240 3406 y Fn(i)p Fl(=0)1369 3290 y Fu(\()p Fr(\000)p Fu(1\))1571 3249 y Fn(m)p Fk(\000)p Fn(i)1716 3150 y Fm(\022)1790 3223 y Fs(k)d Fu(+)d Fs(l)1876 3359 y(i)1995 3150 y Fm(\023)q(\022)2142 3223 y Fs(m)g Fu(+)g Fs(l)j Fr(\000)d Fs(i)h Fr(\000)f Fu(1)2303 3359 y Fs(m)g Fr(\000)h Fs(i)2704 3150 y Fm(\023)2778 3290 y Fs(:)120 3587 y SDict begin H.S end 120 3587 a 120 3587 a SDict begin 17.5 H.A end 120 3587 a 120 3587 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.3) cvn /DEST pdfmark end 120 3587 a Black FA(Lemma)36 b(3.3.)p Black 39 w Fq(F)-7 b(or)32 b(any)g(natur)-5 b(al)32 b(numb)-5 b(ers)32 b Fs(k)s(;)17 b(l)r(;)g(m)32 b Fq(such)g(that)h Fs(l)d Fr(\025)e Fu(1)p Fs(;)17 b(k)30 b(<)e(m)g Fr(\024)g Fs(k)19 b Fu(+)d Fs(l)r Fq(,)33 b(we)120 3733 y(have)1456 3967 y SDict begin H.S end 1456 3967 a 1456 3967 a SDict begin 17.5 H.A end 1456 3967 a 1456 3967 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.4) cvn /DEST pdfmark end 1456 3967 a Fu(\010\()p Fs(k)s(;)17 b(l)r(;)g(k)s Fu(\))28 b(=)f(1)p Fs(;)p Black -1916 w Fu(\(3.4\))p Black 1441 4149 a(\010\()p Fs(k)s(;)17 b(l)r(;)g(m)p Fu(\))28 b(=)g(0)p Fs(:)p Black -1933 w Fu(\(3.5\))p Black Black 120 4388 a Fq(Pr)-5 b(o)g(of.)p Black 41 w Fu(T)d(o)33 b(pro)m(v)m(e)h(the)f (form)m(ula)d(\()1388 4388 y SDict begin H.S end 1388 4388 a Black Fu(3.5)p Black 1513 4324 a SDict begin H.R end 1513 4324 a 1513 4388 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 1513 4388 a Fu(\))i(w)m(e)i(will)c(also)i(need)h(to)g(sho)m(w)g(that)539 4768 y SDict begin H.S end 539 4768 a 539 4768 a SDict begin 17.5 H.A end 539 4768 a 539 4768 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.6) cvn /DEST pdfmark end 539 4768 a Fu(\002\()p Fs(k)25 b Fu(+)d Fs(l)r(;)17 b(k)25 b Fu(+)d Fs(l)r(;)17 b(k)25 b Fu(+)d Fs(l)r(;)17 b(n)23 b Fu(+)f Fs(l)i Fr(\000)f Fu(1)p Fs(;)17 b(n)p Fu(\))27 b(=)h(0)97 b(for)32 b(an)m(y)h Fs(n)28 b Fr(\025)g Fs(k)e Fu(+)c Fs(l)r(:)p Black -2835 w Fu(\(3.6\))p Black 220 5002 a(W)-8 b(e)39 b(will)f(pro)m(v)m(e)i(all)e(form)m(ulas)g(b)m(y)i (induction)e(on)i Fs(l)r Fu(.)64 b(F)-8 b(or)39 b Fs(l)i Fu(=)f(1)f(in)f(form)m(ula)g(\()3227 5002 y SDict begin H.S end 3227 5002 a Black Fu(3.4)p Black 3352 4938 a SDict begin H.R end 3352 4938 a 3352 5002 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 3352 5002 a Fu(\))h(w)m(e)120 5147 y(ha)m(v)m(e)p Black Black eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 232 -8 a SDict begin H.S end 232 -8 a 232 -8 a SDict begin H.R end 232 -8 a 232 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end 232 -8 a Black 240 70 a Fo(8)824 b(ANTONIO)24 b(A)-9 b(VIL)1693 52 y(\023)1686 70 y(ES,)25 b(WITOLD)g(MAR)n(CISZEWSKI)p Black 355 572 a Fu(\010\()p Fs(k)s(;)17 b Fu(1)p Fs(;)g(k)s Fu(\))28 b(=)930 448 y Fn(k)877 477 y Fm(X)892 687 y Fn(i)p Fl(=0)1021 572 y Fu(\()p Fr(\000)p Fu(1\))1223 531 y Fn(k)r Fk(\000)p Fn(i)1345 432 y Fm(\022)1418 505 y Fs(k)e Fu(+)c(1)1513 640 y Fs(i)1641 432 y Fm(\023)q(\022)1788 505 y Fs(k)j Fu(+)d(1)g Fr(\000)h Fs(i)f Fr(\000)h Fu(1)1958 640 y Fs(k)i Fr(\000)e Fs(i)2337 432 y Fm(\023)2438 572 y Fu(=)2594 448 y Fn(k)2541 477 y Fm(X)2556 687 y Fn(i)p Fl(=0)2685 572 y Fu(\()p Fr(\000)p Fu(1\))2887 531 y Fn(k)r Fk(\000)p Fn(i)3009 432 y Fm(\022)3082 505 y Fs(k)i Fu(+)d(1)3177 640 y Fs(i)3305 432 y Fm(\023)761 923 y Fu(=)27 b(1)22 b Fr(\000)h Fu(1)f Fr(\000)1258 798 y Fn(k)1205 828 y Fm(X)1220 1038 y Fn(i)p Fl(=0)1349 923 y Fu(\()p Fr(\000)p Fu(1\))1551 881 y Fn(k)r Fl(+1)p Fk(\000)p Fn(i)1763 782 y Fm(\022)1836 855 y Fs(k)j Fu(+)e(1)1931 991 y Fs(i)2059 782 y Fm(\023)2160 923 y Fu(=)28 b(1)22 b Fr(\000)2442 798 y Fn(k)r Fl(+1)2434 828 y Fm(X)2449 1038 y Fn(i)p Fl(=0)2595 782 y Fm(\022)2668 855 y Fs(k)j Fu(+)d(1)2763 991 y Fs(i)2891 782 y Fm(\023)2965 923 y Fu(\()p Fr(\000)p Fu(1\))3167 881 y Fn(k)r Fl(+1)p Fk(\000)p Fn(i)2469 1180 y Fu(=)28 b(1)22 b Fr(\000)g Fu(\(1)g Fr(\000)h Fu(1\))3039 1139 y Fn(k)r Fl(+1)3199 1180 y Fu(=)28 b(1)p Fs(:)340 1443 y Fu(F)-8 b(or)31 b Fs(l)f Fu(=)e(1)k(the)h(form)m(ula)e (\()1322 1443 y SDict begin H.S end 1322 1443 a Black Fu(3.6)p Black 1446 1378 a SDict begin H.R end 1446 1378 a 1446 1443 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.6) cvn H.B /ANN pdfmark end 1446 1443 a Fu(\))i(has)g(the)g(form)432 1894 y(\002\()p Fs(k)25 b Fu(+)d(1)p Fs(;)17 b(k)25 b Fu(+)d(1)p Fs(;)17 b(k)24 b Fu(+)e(1)p Fs(;)17 b(n)22 b Fu(+)g(1)g Fr(\000)h Fu(1)p Fs(;)17 b(n)p Fu(\))27 b(=)2022 1770 y Fn(k)r Fl(+1)2015 1800 y Fm(X)2029 2010 y Fn(i)p Fl(=0)2159 1894 y Fu(\()p Fr(\000)p Fu(1\))2361 1853 y Fn(k)r Fl(+1)p Fk(\000)p Fn(i)2572 1754 y Fm(\022)2646 1827 y Fs(k)e Fu(+)d(1)2741 1963 y Fs(i)2869 1754 y Fm(\023\022)3015 1827 y Fs(n)h Fr(\000)f Fs(i)3015 1963 y(n)h Fr(\000)f Fs(i)3229 1754 y Fm(\023)1488 2245 y Fu(=)1599 2120 y Fn(k)r Fl(+1)1592 2150 y Fm(X)1606 2360 y Fn(i)p Fl(=0)1752 2104 y Fm(\022)1825 2177 y Fs(k)k Fu(+)c(1)1920 2313 y Fs(i)2048 2104 y Fm(\023)2122 2245 y Fu(\()p Fr(\000)p Fu(1\))2324 2204 y Fn(k)r Fl(+1)p Fk(\000)p Fn(i)2563 2245 y Fu(=)28 b(\(1)22 b Fr(\000)g Fu(1\))2962 2204 y Fn(k)r Fl(+1)3123 2245 y Fu(=)27 b(0)p Fs(:)340 2571 y Fu(F)-8 b(or)21 b Fs(l)30 b Fu(=)d(1)22 b(the)h(only)e(p)s(ossible)g (v)-5 b(alue)22 b(of)f Fs(m)i Fu(is)e Fs(k)t Fu(+)q(1,)i(so)g(for)e(\() 2508 2571 y SDict begin H.S end 2508 2571 a Black Fu(3.5)p Black 2633 2506 a SDict begin H.R end 2633 2506 a 2633 2571 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 2633 2571 a Fu(\))h(w)m(e)h(ha)m(v)m(e)g (\010\()p Fs(k)s(;)17 b Fu(1)p Fs(;)g(k)t Fu(+)q(1\))26 b(=)240 2716 y(\002\()p Fs(k)f Fu(+)d(1)p Fs(;)17 b(k)25 b Fu(+)d(1)p Fs(;)17 b(k)24 b Fu(+)e(1)p Fs(;)17 b(k)25 b Fu(+)d(1)g(+)g(1)g Fr(\000)h Fu(1)p Fs(;)17 b(k)24 b Fu(+)e(1\))28 b(=)f(0.)340 2861 y(T)-8 b(o)33 b(complete)h(the)g (inductiv)m(e)g(step)g(w)m(e)h(will)d(use)i(Lemma)2548 2861 y SDict begin H.S end 2548 2861 a Black Fu(3.2)p Black 2672 2797 a SDict begin H.R end 2672 2797 a 2672 2861 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.2) cvn H.B /ANN pdfmark end 2672 2861 a Fu(.)48 b(By)34 b(\()2939 2861 y SDict begin H.S end 2939 2861 a Black Fu(3.2)p Black 3063 2797 a SDict begin H.R end 3063 2797 a 3063 2861 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 3063 2861 a Fu(\),)h(for)e Fs(k)g Fr(\024)d Fs(p)g Fr(\024)240 3006 y Fs(k)25 b Fu(+)d Fs(l)35 b Fu(w)m(e)f(ha)m(v)m(e)1009 3415 y SDict begin H.S end 1009 3415 a 1009 3415 a SDict begin 17.5 H.A end 1009 3415 a 1009 3415 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.7) cvn /DEST pdfmark end 1009 3415 a Fu(\010\()p Fs(k)s(;)17 b(l)24 b Fu(+)e(1)p Fs(;)17 b(p)p Fu(\))27 b(=)h(\002\()p Fs(p;)17 b(p;)g(k)24 b Fu(+)e Fs(l)j Fu(+)d(1)p Fs(;)17 b(p)k Fu(+)h Fs(l)r(;)17 b(p)p Fu(\))p Black -2485 w(\(3.7\))p Black 1045 3596 a(=)28 b(\002\()p Fs(p;)17 b(p;)g(k)24 b Fu(+)e Fs(l)r(;)17 b(p)22 b Fu(+)g Fs(l)j Fr(\000)d Fu(1)p Fs(;)17 b(p)p Fu(\))27 b(=)h(\010\()p Fs(k)s(;)17 b(l)r(;)g(p)p Fu(\))p Fs(:)340 3859 y Fu(In)27 b(particular,)g(\010\()p Fs(k)s(;)17 b(l)d Fu(+)d(1)p Fs(;)17 b(k)s Fu(\))27 b(=)h(\010\()p Fs(k)s(;)17 b(l)r(;)g(k)s Fu(\).)42 b(F)-8 b(rom)26 b(\()2312 3859 y SDict begin H.S end 2312 3859 a Black Fu(3.3)p Black 2436 3795 a SDict begin H.R end 2436 3795 a 2436 3859 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 2436 3859 a Fu(\))h(w)m(e)i(obtain,)e(for)g Fs(n)h Fr(\025)g Fs(k)15 b Fu(+)c Fs(l)j Fu(+)d(1,)1008 4267 y SDict begin H.S end 1008 4267 a 1008 4267 a SDict begin 17.5 H.A end 1008 4267 a 1008 4267 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.8) cvn /DEST pdfmark end 1008 4267 a Fu(\002\()p Fs(k)25 b Fu(+)d Fs(l)j Fu(+)d(1)p Fs(;)17 b(k)24 b Fu(+)e Fs(l)j Fu(+)d(1)p Fs(;)17 b(k)24 b Fu(+)e Fs(l)j Fu(+)d(1)p Fs(;)17 b(n)22 b Fu(+)g Fs(l)r(;)17 b(n)p Fu(\))p Black -2486 w(\(3.8\))p Black 1163 4449 a(=)28 b Fr(\000)p Fu(\002\()p Fs(k)d Fu(+)d Fs(l)r(;)17 b(k)26 b Fu(+)c Fs(l)r(;)17 b(k)25 b Fu(+)d Fs(l)r(;)17 b(n)22 b Fu(+)g Fs(l)j Fr(\000)d Fu(1)p Fs(;)17 b(n)p Fu(\))340 4711 y(whic)m(h)36 b(completes)f(the)h(pro)s(of)e(of)h(\() 1655 4711 y SDict begin H.S end 1655 4711 a Black Fu(3.6)p Black 1780 4647 a SDict begin H.R end 1780 4647 a 1780 4711 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.6) cvn H.B /ANN pdfmark end 1780 4711 a Fu(\).)52 b(T)-8 b(o)35 b(complete)g(the)h(pro)s(of)f(of)g(\()3039 4711 y SDict begin H.S end 3039 4711 a Black Fu(3.5)p Black 3163 4647 a SDict begin H.R end 3163 4647 a 3163 4711 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 3163 4711 a Fu(\))h(w)m(e)g(should)240 4857 y(consider)31 b(t)m(w)m(o)g(cases.)44 b(If)30 b Fs(m)e Fr(\024)g Fs(k)21 b Fu(+)d Fs(l)32 b Fu(then)f(\010\()p Fs(k)s(;)17 b(l)j Fu(+)e(1)p Fs(;)f(m)p Fu(\))27 b(=)h(\010\()p Fs(k)s(;)17 b(l)r(;)g(m)p Fu(\))30 b(b)m(y)i(\()3144 4857 y SDict begin H.S end 3144 4857 a Black Fu(3.7)p Black 3268 4793 a SDict begin H.R end 3268 4793 a 3268 4857 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.7) cvn H.B /ANN pdfmark end 3268 4857 a Fu(\).)43 b(If)30 b Fs(m)e Fu(=)240 5002 y Fs(k)5 b Fu(+)r Fs(l)t Fu(+)r(1)22 b(then)h(\010\()p Fs(k)s(;)17 b(l)t Fu(+)r(1)p Fs(;)g(k)5 b Fu(+)r Fs(l)t Fu(+)r(1\))26 b(=)h(\002\()p Fs(k)5 b Fu(+)r Fs(l)t Fu(+)r(1)p Fs(;)17 b(k)5 b Fu(+)r Fs(l)t Fu(+)r(1)p Fs(;)17 b(k)5 b Fu(+)r Fs(l)t Fu(+)r(1)p Fs(;)17 b(k)5 b Fu(+)r(2)p Fs(l)t Fu(+)r(1)p Fs(;)17 b(k)5 b Fu(+)r Fs(l)t Fu(+)r(1\))22 b(=)27 b(0)240 5147 y(b)m(y)33 b(\()413 5147 y SDict begin H.S end 413 5147 a Black Fu(3.6)p Black 538 5083 a SDict begin H.R end 538 5083 a 538 5147 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.6) cvn H.B /ANN pdfmark end 538 5147 a Fu(\).)2980 b Fg(\003)p Black Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 112 -8 a SDict begin H.S end 112 -8 a 112 -8 a SDict begin H.R end 112 -8 a 112 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end 112 -8 a Black 442 66 a Fo(EXTENSION)32 b(OPERA)-6 b(TORS)32 b(ON)g(BALLS)h(AND)f(ON)h (SP)-6 b(A)n(CES)33 b(OF)g(FINITE)g(SETS)284 b(9)p Black 220 266 a Fu(F)-8 b(or)36 b(an)m(y)i(natural)e(n)m(um)m(b)s(ers)j Fs(k)s(;)17 b(m;)g(s)37 b Fu(suc)m(h)i(that)e Fs(s)e Fr(\025)i Fs(k)s Fu(,)h(w)m(e)h(put)e Fs(j)42 b Fu(=)36 b(min)o(\()p Fs(k)s(;)17 b(m)p Fu(\))37 b(and)120 411 y(w)m(e)d(de\014ne)928 669 y(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))27 b(=)1539 541 y Fn(j)1484 575 y Fm(X)1498 784 y Fn(i)p Fl(=0)1627 669 y Fu(\()p Fr(\000)p Fu(1\))1829 628 y Fn(k)r Fk(\000)p Fn(i)1951 529 y Fm(\022)2025 602 y Fs(m)2051 738 y(i)2110 529 y Fm(\023\022)2261 602 y Fs(s)22 b Fr(\000)g Fs(i)2257 738 y(k)j Fr(\000)d Fs(i)2466 529 y Fm(\023)2539 669 y Fs(:)120 954 y SDict begin H.S end 120 954 a 120 954 a SDict begin 17.5 H.A end 120 954 a 120 954 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.4) cvn /DEST pdfmark end 120 954 a Black FA(Lemma)39 b(3.4.)p Black 42 w Fq(F)-7 b(or)34 b(any)h(natur)-5 b(al)35 b(numb)-5 b(ers)34 b Fs(k)s(;)17 b(m;)g(s)35 b Fq(such)f(that)i Fs(s)27 b Fr(\025)h Fs(k)s Fq(,)35 b(we)g(have)836 1194 y SDict begin H.S end 836 1194 a 836 1194 a SDict begin 17.5 H.A end 836 1194 a 836 1194 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.9) cvn /DEST pdfmark end 836 1194 a Fu(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))83 b(=)g(\()p Fr(\000)p Fu(1\))1705 1153 y Fn(k)1748 1053 y Fm(\022)1821 1126 y Fs(s)22 b Fr(\000)h Fs(m)1920 1262 y(k)2074 1053 y Fm(\023)2247 1194 y Fq(if)35 b Fs(s)27 b Fr(\025)h Fs(k)e Fu(+)c Fs(m;)p Black -2687 w Fu(\(3.9\))p Black 836 1408 a(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))83 b(=)g(0)99 b Fq(if)35 b Fs(k)c Fr(\025)d Fu(1)p Fs(;)17 b(m)27 b Fr(\024)h Fs(s)g(<)g(k)d Fu(+)d Fs(m)p Black -2560 w Fu(\(3.10\))p Black Black 120 1635 a Fq(Pr)-5 b(o)g(of.)p Black 41 w Fu(W)d(e)44 b(will)d(start)i(with)g(the)h(form)m (ula)e(\()1855 1635 y SDict begin H.S end 1855 1635 a Black Fu(3.9)p Black 1979 1571 a SDict begin H.R end 1979 1571 a 1979 1635 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 1979 1635 a Fu(\))h(and)h(use)g (the)g(induction)e(on)i Fs(m)p Fu(.)76 b(F)-8 b(or)120 1780 y Fs(m)35 b Fu(=)f(0,)k Fs(j)i Fu(=)35 b(0)h(and)h(our)f(form)m (ula)f(ob)m(viously)i(holds)f(true.)56 b(Assume)38 b(that)e(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))34 b(=)120 1926 y(\()p Fr(\000)p Fu(1\))322 1890 y Fn(k)365 1845 y Fm(\000)410 1881 y Fn(s)p Fk(\000)p Fn(m)466 1960 y(k)560 1845 y Fm(\001)646 1926 y Fu(for)39 b(an)m(y)i Fs(k)s(;)17 b(s)39 b Fu(suc)m(h)j(that)d Fs(s)h Fr(\025)h Fs(k)30 b Fu(+)d Fs(m)p Fu(.)66 b(T)-8 b(o)40 b(pro)m(v)m(e)h(the)f(inductiv)m(e)g(step)h(w)m(e)120 2071 y(shall)33 b(consider)i(t)m(w)m(o)g(cases.)50 b(If)34 b Fs(k)g Fr(\024)d Fs(m)j Fu(then)h(min)o(\()p Fs(k)s(;)17 b(m)p Fu(\))30 b(=)h(min)n(\()p Fs(k)s(;)17 b(m)24 b Fu(+)f(1\))30 b(=)h Fs(k)s Fu(,)k(and)f(b)m(y)120 2216 y(\()158 2216 y SDict begin H.S end 158 2216 a Black Fu(3.2)p Black 283 2152 a SDict begin H.R end 283 2152 a 283 2216 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 283 2216 a Fu(\))658 2429 y SDict begin H.S end 658 2429 a 658 2429 a SDict begin 17.5 H.A end 658 2429 a 658 2429 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.11) cvn /DEST pdfmark end 658 2429 a Fu(\011\()p Fs(k)s(;)17 b(m)22 b Fu(+)g(1)p Fs(;)17 b(s)p Fu(\))27 b(=)g(\002\()p Fs(k)s(;)17 b(k)s(;)g(m)22 b Fu(+)g(1)p Fs(;)17 b(s;)g(k)s Fu(\))27 b(=)h(\002\()p Fs(k)s(;)17 b(k)s(;)g(m;)g(s)k Fr(\000)i Fu(1)p Fs(;)17 b(k)s Fu(\))p Black -2974 w(\(3.11\))p Black 401 2643 a(=)27 b(\011\()p Fs(k)s(;)17 b(m;)g(s)22 b Fr(\000)h Fu(1\))k(=)h(\()p Fr(\000)p Fu(1\))1433 2602 y Fn(k)1475 2503 y Fm(\022)1549 2576 y Fs(s)22 b Fr(\000)g Fu(1)g Fr(\000)h Fs(m)1733 2711 y(k)1972 2503 y Fm(\023)2073 2643 y Fu(=)28 b(\()p Fr(\000)p Fu(1\))2379 2602 y Fn(k)2422 2503 y Fm(\022)2495 2576 y Fs(s)22 b Fr(\000)h Fu(\()p Fs(m)f Fu(+)g(1\))2717 2711 y Fs(k)2993 2503 y Fm(\023)3066 2643 y Fs(:)120 2888 y Fu(If)33 b Fs(k)d(>)e(m)33 b Fu(then)g(min)n(\()p Fs(k)s(;)17 b(m)p Fu(\))28 b(=)g(min)n(\()p Fs(k)s(;)17 b(m)22 b Fu(+)g(1\))28 b(=)f Fs(m)p Fu(,)33 b(and)g(b)m(y)g(\()2525 2888 y SDict begin H.S end 2525 2888 a Black Fu(3.1)p Black 2650 2824 a SDict begin H.R end 2650 2824 a 2650 2888 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 2650 2888 a Fu(\))f(and)h(\()2948 2888 y SDict begin H.S end 2948 2888 a Black Fu(3.3)p Black 3073 2824 a SDict begin H.R end 3073 2824 a 3073 2888 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 3073 2888 a Fu(\))1379 3101 y SDict begin H.S end 1379 3101 a 1379 3101 a SDict begin 17.5 H.A end 1379 3101 a 1379 3101 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.12) cvn /DEST pdfmark end 1379 3101 a Fu(\011\()p Fs(k)s(;)17 b(m)22 b Fu(+)g(1)p Fs(;)17 b(s)p Fu(\))28 b(=)f(\002\()p Fs(m)22 b Fu(+)g(1)p Fs(;)17 b(k)s(;)g(m)22 b Fu(+)g(1)p Fs(;)17 b(s;)g(k)s Fu(\))p Black -2974 w(\(3.12\))p Black 801 3283 a(=)28 b Fr(\000)p Fu(\002\()p Fs(m)22 b Fu(+)g(1)p Fs(;)17 b(k)25 b Fu(+)d(1)p Fs(;)17 b(m)22 b Fu(+)g(1)p Fs(;)17 b(s;)g(k)s Fu(\))27 b(=)g(\002\()p Fs(m;)17 b(k)s(;)g(m;)g(s)22 b Fr(\000)h Fu(1)p Fs(;)17 b(k)s Fu(\))401 3497 y(=)27 b(\011\()p Fs(k)s(;)17 b(m;)g(s)22 b Fr(\000)h Fu(1\))k(=)h(\()p Fr(\000)p Fu(1\))1433 3455 y Fn(k)1475 3356 y Fm(\022)1549 3429 y Fs(s)22 b Fr(\000)g Fu(1)g Fr(\000)h Fs(m)1733 3565 y(k)1972 3356 y Fm(\023)2073 3497 y Fu(=)28 b(\()p Fr(\000)p Fu(1\))2379 3455 y Fn(k)2422 3356 y Fm(\022)2495 3429 y Fs(s)22 b Fr(\000)h Fu(\()p Fs(m)f Fu(+)g(1\))2717 3565 y Fs(k)2993 3356 y Fm(\023)3066 3497 y Fs(:)220 3742 y Fu(W)-8 b(e)33 b(will)d(split)h(the)i(pro)s(of)f(of)g(\()1361 3742 y SDict begin H.S end 1361 3742 a Black Fu(3.10)p Black 1534 3677 a SDict begin H.R end 1534 3677 a 1534 3742 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 1534 3742 a Fu(\))h(in)m(to)f(t)m(w)m(o)h (cases.)220 3887 y(If)44 b Fs(k)50 b Fr(\024)e Fs(m)d Fu(then)g(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))47 b(=)g(\010\()p Fs(m)31 b Fr(\000)f Fs(s)g Fu(+)g Fs(k)j Fr(\000)e Fu(1)p Fs(;)17 b(s)30 b Fr(\000)g Fs(k)j Fu(+)d(1)p Fs(;)17 b(k)s Fu(\).)78 b(Observ)m(e)46 b(that)120 4032 y Fs(s)21 b Fr(\000)g Fs(k)j Fu(+)d(1)28 b Fr(\025)g Fu(1,)k Fs(m)c Fr(\025)g Fs(k)j(>)c(m)21 b Fr(\000)h Fs(s)f Fu(+)g Fs(k)j Fr(\000)d Fu(1)32 b(since)g Fs(k)f Fr(\024)d Fs(s)p Fu(,)k Fs(k)f Fr(\024)d Fs(m)k Fu(and)g Fs(m)c Fr(\024)h Fs(s)p Fu(.)43 b(Therefore)120 4178 y(b)m(y)33 b(form)m(ula)e(\()651 4178 y SDict begin H.S end 651 4178 a Black Fu(3.5)p Black 776 4113 a SDict begin H.R end 776 4113 a 776 4178 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 776 4178 a Fu(\))h(from)g(Lemma)1425 4178 y SDict begin H.S end 1425 4178 a Black Fu(3.3)p Black 1549 4113 a SDict begin H.R end 1549 4113 a 1549 4178 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.3) cvn H.B /ANN pdfmark end 1549 4178 a 33 w Fu(w)m(e)h(ha)m(v)m(e)h(\010\()p Fs(m)23 b Fr(\000)g Fs(s)f Fu(+)g Fs(k)j Fr(\000)e Fu(1)p Fs(;)17 b(s)k Fr(\000)i Fs(k)i Fu(+)d(1)p Fs(;)17 b(k)s Fu(\))27 b(=)h(0.)220 4323 y(If)23 b Fs(k)31 b(>)d(m)c Fu(then)g(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))27 b(=)1372 4248 y Fm(P)1477 4274 y Fn(m)1477 4352 y(i)p Fl(=0)1595 4323 y Fu(\()p Fr(\000)p Fu(1\))1797 4287 y Fn(k)r Fk(\000)p Fn(i)1919 4242 y Fm(\000)1965 4279 y Fn(m)1984 4357 y(i)2027 4242 y Fm(\001\000)2121 4279 y Fn(s)p Fk(\000)p Fn(i)2118 4357 y(k)r Fk(\000)p Fn(i)2236 4242 y Fm(\001)2305 4323 y Fu(and)d(w)m(e)h(will)c(pro)m(v)m(e)k(our)f(form)m(ula)120 4468 y(b)m(y)33 b(induction)f(on)h Fs(s)22 b Fr(\000)g Fs(k)s Fu(.)44 b(Observ)m(e)34 b(that)f Fs(m)28 b Fr(\025)g Fu(1)k(since)h Fs(k)e Fr(\024)d Fs(s)g(<)f(k)e Fu(+)d Fs(m)p Fu(.)44 b(If)33 b Fs(s)27 b Fu(=)h Fs(k)35 b Fu(then)671 4726 y(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))27 b(=)1268 4602 y Fn(m)1227 4632 y Fm(X)1242 4842 y Fn(i)p Fl(=0)1371 4726 y Fu(\()p Fr(\000)p Fu(1\))1573 4685 y Fn(k)r Fk(\000)p Fn(i)1695 4586 y Fm(\022)1768 4659 y Fs(m)1794 4795 y(i)1853 4586 y Fm(\023)1954 4726 y Fu(=)h(\()p Fr(\000)p Fu(1\))2260 4685 y Fn(k)r Fk(\000)p Fn(m)2420 4726 y Fu(\(1)22 b Fr(\000)g Fu(1\))2715 4685 y Fn(m)2809 4726 y Fu(=)28 b(0)p Fs(:)120 5002 y Fu(The)36 b(inductiv)m(e)f(step)h(follo)m(ws)d (from)h(the)h(form)m(ula)e(\011\()p Fs(k)s(;)17 b(m;)g(s)p Fu(\))31 b(=)g(\011\()p Fs(k)s(;)17 b(m)24 b Fr(\000)g Fu(1)p Fs(;)17 b(s)23 b Fr(\000)i Fu(1\),)35 b(cf.)120 5147 y(\()158 5147 y SDict begin H.S end 158 5147 a Black Fu(3.12)p Black 331 5083 a SDict begin H.R end 331 5083 a 331 5147 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.12) cvn H.B /ANN pdfmark end 331 5147 a Fu(\).)3067 b Fg(\003)p Black Black eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 232 -8 a SDict begin H.S end 232 -8 a 232 -8 a SDict begin H.R end 232 -8 a 232 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end 232 -8 a Black 240 70 a Fo(10)786 b(ANTONIO)24 b(A)-9 b(VIL)1693 52 y(\023)1686 70 y(ES,)25 b(WITOLD)g(MAR)n(CISZEWSKI)p Black 340 266 a Fu(The)33 b(next)h(lemma)c(sho)m(ws)k(that)e(the)i (function)e Fs(T)2173 281 y Fn(X)2273 266 y Fu(of)g(Theorem)2796 266 y SDict begin H.S end 2796 266 a Black Fu(3.1)p Black 2920 201 a SDict begin H.R end 2920 201 a 2920 266 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.1) cvn H.B /ANN pdfmark end 2920 266 a 33 w Fu(is)g(w)m(ell)g(de\014ned.)240 488 y SDict begin H.S end 240 488 a 240 488 a SDict begin 17.5 H.A end 240 488 a 240 488 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.5) cvn /DEST pdfmark end 240 488 a Black FA(Lemma)47 b(3.5.)p Black 45 w Fq(F)-7 b(or)40 b(any)h(set)g Fs(X)8 b Fq(,)42 b(p)-5 b(ositive)40 b(inte)-5 b(gers)40 b Fs(m;)17 b(n)p Fq(,)43 b Fs(m)c(<)g(n)p Fq(,)j(and)e(a)h(c) -5 b(ontinuous)240 633 y(function)25 b Fs(f)38 b Fr(2)29 b Fs(C)7 b Fu(\()p Fs(\033)965 648 y Fn(m)1031 633 y Fu(\(2)1118 597 y Fn(X)1186 633 y Fu(\)\))p Fq(,)27 b(the)e(function)g Fs(T)1902 648 y Fn(X)1970 633 y Fu(\()p Fs(f)11 b Fu(\))25 b Fq(de\014ne)-5 b(d)24 b(in)h(The)-5 b(or)g(em)2954 633 y SDict begin H.S end 2954 633 a Black Fq(3.1)p Black 3084 568 a SDict begin H.R end 3084 568 a 3084 633 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.1) cvn H.B /ANN pdfmark end 3084 633 a 25 w Fq(is)25 b(c)-5 b(ontinuous)240 778 y(on)34 b Fs(\033)434 793 y Fn(n)482 778 y Fu(\(2)569 742 y Fn(X)636 778 y Fu(\))p Fq(.)p Black 240 1000 a(Pr)-5 b(o)g(of.)p Black 41 w Fu(The)30 b(space)f Fs(\033)1046 1015 y Fn(n)1094 1000 y Fu(\(2)1181 964 y Fn(X)1248 1000 y Fu(\),)g(b)s(eing)f(Eb)s(erlein)g(compact,)h(is)f(a)g(F)-8 b(r)m(\023)-46 b(ec)m(het)30 b(top)s(ological)25 b(space.)240 1145 y(Therefore,)i(it)c(is)g(enough)h(to)g(sho)m(w)h(that,)g(for)e(ev) m(ery)j(sequence)h(\()p Fs(\037)2666 1160 y Fn(A)2719 1172 y Fh(k)2761 1145 y Fu(\))c(of)h(p)s(oin)m(ts)f(of)g Fs(\033)3366 1160 y Fn(n)3414 1145 y Fu(\(2)3501 1109 y Fn(X)3568 1145 y Fu(\))t Fr(n)240 1291 y Fs(\033)295 1306 y Fn(m)362 1291 y Fu(\(2)449 1255 y Fn(X)516 1291 y Fu(\),)36 b(con)m(v)m(erging)f(to)g(a)g(p)s(oin)m(t)f Fs(\037)1628 1306 y Fn(B)1720 1291 y Fr(2)f Fs(\033)1874 1306 y Fn(n)1921 1291 y Fu(\(2)2008 1255 y Fn(X)2075 1291 y Fu(\))i(w)m(e)h(ha)m(v)m(e)h Fs(T)2579 1306 y Fn(X)2646 1291 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)2880 1306 y Fn(A)2933 1318 y Fh(k)2975 1291 y Fu(\))32 b Fr(!)f Fs(T)3233 1306 y Fn(X)3301 1291 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)3535 1306 y Fn(B)3595 1291 y Fu(\).)240 1436 y(Without)31 b(loss)g(of)h(generalit)m(y)f(w)m(e)h(ma)m(y)g(assume)g (that)g Fs(B)g Fr(\032)d Fs(A)2561 1451 y Fn(k)2635 1436 y Fu(for)i(all)f Fs(k)s Fu(,)i(and)g(all)d(sets)k Fs(A)3617 1451 y Fn(k)240 1581 y Fu(ha)m(v)m(e)40 b(the)g(same)f(cardinalit)m(y)e Fs(r)s Fu(.)63 b(One)40 b(can)f(easily)g(v)m(erify)g(that,)i(for)e(ev)m (ery)i Fs(D)g Fr(\032)f Fs(B)k Fu(and)240 1727 y Fs(C)310 1742 y Fn(k)393 1727 y Fr(\032)e Fu(\()p Fs(A)623 1742 y Fn(k)693 1727 y Fr(n)27 b Fs(B)5 b Fu(\),)43 b(w)m(e)e(ha)m(v)m(e)h Fs(\037)1402 1742 y Fn(C)1452 1754 y Fh(k)1491 1742 y Fk([)p Fn(D)1643 1727 y Fr(!)e Fs(\037)1844 1742 y Fn(D)1909 1727 y Fu(.)67 b(Let)40 b Fs(p)h Fu(=)g Fr(j)p Fs(B)5 b Fr(j)p Fu(.)67 b(F)-8 b(or)39 b(an)m(y)i Fs(D)j Fr(2)d Fu([)p Fs(B)5 b Fu(])3360 1690 y Fk(\024)p Fn(m)3482 1727 y Fu(,)43 b(let)240 1872 y Fs(j)280 1887 y Fn(D)372 1872 y Fu(=)28 b(min)n(\()p Fs(m)s Fr(\000)s(j)p Fs(D)s Fr(j)p Fs(;)17 b(r)6 b Fr(\000)s Fs(p)p Fu(\).)42 b(W)-8 b(e)24 b(shall)e(consider)i(t)m(w)m(o)g(cases.)42 b(If)23 b Fs(p)28 b Fr(\024)g Fs(m)p Fu(,)d(i.e.,)g Fs(\037)3163 1887 y Fn(B)3252 1872 y Fr(2)j Fs(\033)3401 1887 y Fn(m)3468 1872 y Fu(\(2)3555 1836 y Fn(X)3622 1872 y Fu(\))240 2017 y(then)33 b(w)m(e)h(ha)m(v)m(e)240 2228 y Fs(T)297 2243 y Fn(X)365 2228 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)599 2243 y Fn(A)652 2255 y Fh(k)693 2228 y Fu(\))135 b(=)1193 2133 y Fm(X)1077 2353 y Fn(B)1130 2365 y Fh(k)1168 2353 y Fk(2)p Fl([)p Fn(A)1288 2365 y Fh(k)1326 2353 y Fl(])1346 2334 y Fi(\024)p Fh(m)1452 2228 y Fu(\()p Fr(\000)p Fu(1\))1654 2187 y Fn(m)p Fk(\000j)p Fn(B)1844 2199 y Fh(k)1882 2187 y Fk(j)1906 2087 y Fm(\022)1980 2160 y Fs(r)24 b Fr(\000)f(j)p Fs(B)2250 2175 y Fn(k)2292 2160 y Fr(j)f(\000)h Fu(1)2046 2296 y Fs(m)f Fr(\000)h(j)p Fs(B)2355 2311 y Fn(k)2397 2296 y Fr(j)2491 2087 y Fm(\023)2564 2228 y Fs(f)11 b Fu(\()p Fs(\037)2722 2243 y Fn(B)2775 2255 y Fh(k)2817 2228 y Fu(\))866 2569 y(=)1277 2474 y Fm(X)1077 2694 y Fn(C)1127 2706 y Fh(k)1165 2694 y Fk(2)p Fl([)p Fn(A)1285 2706 y Fh(k)1323 2694 y Fk(n)p Fn(B)s Fl(])1434 2675 y Fi(\024)p Fh(m)p Fi(\000)p Fh(p)1621 2569 y Fu(\()p Fr(\000)p Fu(1\))1823 2528 y Fn(m)p Fk(\000j)p Fn(C)2010 2540 y Fh(k)2048 2528 y Fk(j\000)p Fn(p)2163 2428 y Fm(\022)2236 2501 y Fs(r)25 b Fr(\000)d(j)p Fs(C)2502 2516 y Fn(k)2544 2501 y Fr(j)g(\000)h Fs(p)f Fr(\000)h Fu(1)2302 2637 y Fs(m)f Fr(\000)h(j)p Fs(C)2607 2652 y Fn(k)2649 2637 y Fr(j)f(\000)g Fs(p)2913 2428 y Fm(\023)2987 2569 y Fs(f)11 b Fu(\()p Fs(\037)3145 2584 y Fn(C)3195 2596 y Fh(k)3233 2584 y Fk([)p Fn(B)3341 2569 y Fu(\))866 2910 y(+)1089 2815 y Fm(X)1077 3034 y Fn(D)r Ff($)p Fn(B)1491 2815 y Fm(X)1262 3038 y Fn(E)1314 3050 y Fh(k)1351 3038 y Fk(2)p Fl([)p Fn(A)1471 3050 y Fh(k)1509 3038 y Fk(n)p Fn(B)s Fl(])1620 3018 y Fi(\024)p Fh(m)p Fi(\000j)p Fh(D)r Fi(j)1864 2910 y Fu(\()p Fr(\000)p Fu(1\))2066 2869 y Fn(m)p Fk(\000j)p Fn(E)2255 2881 y Fh(k)2293 2869 y Fk(j\000j)p Fn(D)r Fk(j)2471 2769 y Fm(\022)2544 2842 y Fs(r)25 b Fr(\000)d(j)p Fs(E)2812 2857 y Fn(k)2855 2842 y Fr(j)g(\000)g(j)p Fs(D)s Fr(j)g(\000)g Fu(1)2610 2978 y Fs(m)g Fr(\000)h(j)p Fs(E)2917 2993 y Fn(k)2960 2978 y Fr(j)e(\000)i(j)p Fs(D)s Fr(j)3314 2769 y Fm(\023)3387 2910 y Fs(f)11 b Fu(\()p Fs(\037)3545 2925 y Fn(E)3597 2937 y Fh(k)3635 2925 y Fk([)p Fn(D)3746 2910 y Fu(\))814 3231 y Fn(k)r Fk(!1)824 3288 y Fr(\000)-16 b(!)92 b Fs(f)11 b Fu(\()p Fs(\037)1235 3303 y Fn(B)1295 3288 y Fu(\))1350 3159 y Fn(m)p Fk(\000)p Fn(p)1354 3193 y Fm(X)1369 3403 y Fn(i)p Fl(=0)1503 3288 y Fu(\()p Fr(\000)p Fu(1\))1705 3247 y Fn(m)p Fk(\000)p Fn(p)p Fk(\000)p Fn(i)1941 3147 y Fm(\022)2014 3221 y Fs(r)25 b Fr(\000)e Fs(p)2106 3356 y(i)2231 3147 y Fm(\023)q(\022)2378 3221 y Fs(r)i Fr(\000)e Fs(p)f Fr(\000)g Fs(i)h Fr(\000)f Fu(1)2444 3356 y Fs(m)h Fr(\000)f Fs(p)g Fr(\000)h Fs(i)2921 3147 y Fm(\023)866 3640 y Fu(+)1089 3546 y Fm(X)1077 3764 y Fn(D)r Ff($)p Fn(B)1262 3640 y Fs(f)11 b Fu(\()p Fs(\037)1420 3655 y Fn(D)1483 3640 y Fu(\))1568 3512 y Fn(j)1597 3523 y Fh(D)1538 3546 y Fm(X)1553 3756 y Fn(i)p Fl(=0)1682 3640 y Fu(\()p Fr(\000)p Fu(1\))1884 3599 y Fn(m)p Fk(\000j)p Fn(D)r Fk(j\000)p Fn(i)2183 3500 y Fm(\022)2257 3573 y Fs(r)24 b Fr(\000)f Fs(p)2349 3709 y(i)2474 3500 y Fm(\023\022)2621 3573 y Fs(r)i Fr(\000)d(j)p Fs(D)s Fr(j)f(\000)i Fs(i)f Fr(\000)h Fu(1)2687 3709 y Fs(m)f Fr(\000)h(j)p Fs(D)s Fr(j)e(\000)i Fs(i)3254 3500 y Fm(\023)3327 3640 y Fs(:)240 3928 y Fu(By)33 b(Lemma)741 3928 y SDict begin H.S end 741 3928 a Black Fu(3.3)p Black 866 3863 a SDict begin H.R end 866 3863 a 866 3928 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.3) cvn H.B /ANN pdfmark end 866 3928 a 376 4063 a Fn(m)p Fk(\000)p Fn(p)380 4097 y Fm(X)395 4307 y Fn(i)p Fl(=0)529 4192 y Fu(\()p Fr(\000)p Fu(1\))731 4151 y Fn(m)p Fk(\000)p Fn(p)p Fk(\000)p Fn(i)967 4051 y Fm(\022)1040 4124 y Fs(r)25 b Fr(\000)d Fs(p)1132 4260 y(i)1257 4051 y Fm(\023)q(\022)1404 4124 y Fs(r)j Fr(\000)d Fs(p)h Fr(\000)f Fs(i)h Fr(\000)f Fu(1)1470 4260 y Fs(m)g Fr(\000)h Fs(p)f Fr(\000)h Fs(i)1947 4051 y Fm(\023)2048 4192 y Fu(=)28 b(\010\()p Fs(m)22 b Fr(\000)h Fs(p;)17 b(r)24 b Fr(\000)f Fs(m;)17 b(m)23 b Fr(\000)f Fs(p)p Fu(\))28 b(=)f(1)p Fs(:)240 4462 y Fu(F)-8 b(or)37 b(an)m(y)h Fs(D)g Fe($)e Fs(B)5 b Fu(,)39 b Fs(m)26 b Fr(\000)g(j)p Fs(D)s Fr(j)34 b(\025)j Fu(1)g(and)g Fs(r)28 b Fr(\000)e Fs(p)36 b(<)g(r)28 b Fr(\000)d(j)p Fs(D)s Fr(j)g(\000)h Fu(1)35 b Fs(<)h Fu(\()p Fs(m)26 b Fr(\000)g(j)p Fs(D)s Fr(j)p Fu(\))e(+)h(\()p Fs(r)j Fr(\000)e Fs(p)p Fu(\).)240 4608 y(Hence,)34 b(b)m(y)f(form)m(ula)e(\() 1088 4608 y SDict begin H.S end 1088 4608 a Black Fu(3.10)p Black 1262 4543 a SDict begin H.R end 1262 4543 a 1262 4608 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 1262 4608 a Fu(\))h(from)f(Lemma)1910 4608 y SDict begin H.S end 1910 4608 a Black Fu(3.4)p Black 2035 4543 a SDict begin H.R end 2035 4543 a 2035 4608 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.4) cvn H.B /ANN pdfmark end 2035 4608 a 270 4753 a Fn(j)299 4764 y Fh(D)240 4787 y Fm(X)255 4996 y Fn(i)p Fl(=0)384 4881 y Fu(\()p Fr(\000)p Fu(1\))586 4840 y Fn(m)p Fk(\000j)p Fn(D)r Fk(j\000)p Fn(i)885 4741 y Fm(\022)959 4814 y Fs(r)25 b Fr(\000)d Fs(p)1051 4950 y(i)1176 4741 y Fm(\023)q(\022)1323 4814 y Fs(r)j Fr(\000)d(j)p Fs(D)s Fr(j)g(\000)g Fs(i)h Fr(\000)f Fu(1)1389 4950 y Fs(m)g Fr(\000)h(j)p Fs(D)s Fr(j)e(\000)i Fs(i)1956 4741 y Fm(\023)2057 4881 y Fu(=)28 b(\011\()p Fs(m)22 b Fr(\000)g(j)p Fs(D)s Fr(j)p Fs(;)17 b(r)24 b Fr(\000)f Fs(p;)17 b(r)24 b Fr(\000)f(j)p Fs(D)s Fr(j)e(\000)i Fu(1\))k(=)g(0)p Fs(:)240 5147 y Fu(Therefore,)34 b Fs(T)766 5162 y Fn(X)833 5147 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)1067 5162 y Fn(A)1120 5174 y Fh(k)1162 5147 y Fu(\))28 b Fr(\000)-16 b(!)27 b Fs(f)11 b Fu(\()p Fs(\037)1574 5162 y Fn(B)1634 5147 y Fu(\).)p Black Black eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 112 -8 a SDict begin H.S end 112 -8 a 112 -8 a SDict begin H.R end 112 -8 a 112 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.11) cvn /DEST pdfmark end 112 -8 a Black 442 66 a Fo(EXTENSION)32 b(OPERA)-6 b(TORS)32 b(ON)g(BALLS)h(AND)f(ON)h (SP)-6 b(A)n(CES)33 b(OF)g(FINITE)g(SETS)245 b(11)p Black 220 266 a Fu(In)33 b(the)g(second)g(case,)h(when)g Fs(p)27 b(>)h(m)p Fu(,)33 b(w)m(e)g(ha)m(v)m(e)120 504 y Fs(T)177 519 y Fn(X)245 504 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)479 519 y Fn(A)532 531 y Fh(k)573 504 y Fu(\))135 b(=)1073 409 y Fm(X)957 629 y Fn(B)1010 641 y Fh(k)1048 629 y Fk(2)p Fl([)p Fn(A)1168 641 y Fh(k)1206 629 y Fl(])1226 610 y Fi(\024)p Fh(m)1332 504 y Fu(\()p Fr(\000)p Fu(1\))1534 463 y Fn(m)p Fk(\000j)p Fn(B)1724 475 y Fh(k)1762 463 y Fk(j)1786 363 y Fm(\022)1860 436 y Fs(r)24 b Fr(\000)f(j)p Fs(B)2130 451 y Fn(k)2172 436 y Fr(j)f(\000)h Fu(1)1926 572 y Fs(m)f Fr(\000)h(j)p Fs(B)2235 587 y Fn(k)2277 572 y Fr(j)2371 363 y Fm(\023)2444 504 y Fs(f)11 b Fu(\()p Fs(\037)2602 519 y Fn(B)2655 531 y Fh(k)2697 504 y Fu(\))746 845 y(=)1040 750 y Fm(X)957 970 y Fn(D)r Fk(2)p Fl([)p Fn(B)s Fl(])1160 951 y Fi(\024)p Fh(m)1512 750 y Fm(X)1283 973 y Fn(E)1335 985 y Fh(k)1373 973 y Fk(2)p Fl([)p Fn(A)1493 985 y Fh(k)1530 973 y Fk(n)p Fn(B)s Fl(])1641 953 y Fi(\024)p Fh(m)p Fi(\000j)p Fh(D)r Fi(j)1885 845 y Fu(\()p Fr(\000)p Fu(1\))2087 804 y Fn(m)p Fk(\000j)p Fn(E)2276 816 y Fh(k)2314 804 y Fk(j\000j)p Fn(D)r Fk(j)2492 704 y Fm(\022)2565 777 y Fs(r)25 b Fr(\000)e(j)p Fs(E)2834 792 y Fn(k)2876 777 y Fr(j)f(\000)h(j)p Fs(D)s Fr(j)e(\000)h Fu(1)2631 913 y Fs(m)h Fr(\000)f(j)p Fs(E)2938 928 y Fn(k)2981 913 y Fr(j)g(\000)g(j)p Fs(D)s Fr(j)3335 704 y Fm(\023)3409 845 y Fs(f)11 b Fu(\()p Fs(\037)3567 860 y Fn(E)3619 872 y Fh(k)3656 860 y Fk([)p Fn(D)3767 845 y Fu(\))694 1170 y Fn(k)r Fk(!1)704 1227 y Fr(\000)-16 b(!)1040 1132 y Fm(X)957 1352 y Fn(D)r Fk(2)p Fl([)p Fn(B)s Fl(])1160 1333 y Fi(\024)p Fh(m)1283 1227 y Fs(f)11 b Fu(\()p Fs(\037)1441 1242 y Fn(D)1505 1227 y Fu(\))1589 1098 y Fn(j)1618 1109 y Fh(D)1559 1132 y Fm(X)1574 1342 y Fn(i)p Fl(=0)1703 1227 y Fu(\()p Fr(\000)p Fu(1\))1905 1186 y Fn(m)p Fk(\000j)p Fn(D)r Fk(j\000)p Fn(i)2205 1086 y Fm(\022)2278 1159 y Fs(r)25 b Fr(\000)d Fs(p)2370 1295 y(i)2495 1086 y Fm(\023)q(\022)2642 1159 y Fs(r)j Fr(\000)d(j)p Fs(D)s Fr(j)g(\000)g Fs(i)h Fr(\000)f Fu(1)2708 1295 y Fs(m)g Fr(\000)h(j)p Fs(D)s Fr(j)e(\000)i Fs(i)3275 1086 y Fm(\023)3348 1227 y Fs(:)120 1533 y Fu(F)-8 b(or)32 b(an)m(y)i Fs(D)e Fr(2)d Fu([)p Fs(B)5 b Fu(])821 1497 y Fk(\024)p Fn(m)942 1533 y Fu(,)34 b(\()p Fs(m)23 b Fr(\000)g(j)p Fs(D)s Fr(j)p Fu(\))e(+)i(\()p Fs(r)i Fr(\000)e Fs(p)p Fu(\))28 b Fr(\024)i Fs(r)25 b Fr(\000)e(j)p Fs(D)s Fr(j)e(\000)i Fu(1.)45 b(Hence,)35 b(b)m(y)f(form)m(ula)d(\()3377 1533 y SDict begin H.S end 3377 1533 a Black Fu(3.9)p Black 3502 1469 a SDict begin H.R end 3502 1469 a 3502 1533 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 3502 1533 a Fu(\))120 1679 y(from)g(Lemma)699 1679 y SDict begin H.S end 699 1679 a Black Fu(3.4)p Black 823 1615 a SDict begin H.R end 823 1615 a 823 1679 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.4) cvn H.B /ANN pdfmark end 823 1679 a 176 1807 a Fn(j)205 1818 y Fh(D)146 1841 y Fm(X)161 2051 y Fn(i)p Fl(=0)290 1936 y Fu(\()p Fr(\000)p Fu(1\))492 1894 y Fn(m)p Fk(\000j)p Fn(D)r Fk(j\000)p Fn(i)792 1795 y Fm(\022)865 1868 y Fs(r)25 b Fr(\000)e Fs(p)957 2004 y(i)1083 1795 y Fm(\023\022)1229 1868 y Fs(r)i Fr(\000)e(j)p Fs(D)s Fr(j)e(\000)h Fs(i)h Fr(\000)g Fu(1)1295 2004 y Fs(m)g Fr(\000)f(j)p Fs(D)s Fr(j)f(\000)i Fs(i)1862 1795 y Fm(\023)1963 1936 y Fu(=)28 b(\011\()p Fs(m)22 b Fr(\000)h(j)p Fs(D)s Fr(j)p Fs(;)17 b(r)23 b Fr(\000)g Fs(p;)17 b(r)24 b Fr(\000)f(j)p Fs(D)s Fr(j)e(\000)i Fu(1\))901 2250 y(=)k(\()p Fr(\000)p Fu(1\))1206 2209 y Fn(m)p Fk(\000j)p Fn(D)r Fk(j)1427 2109 y Fm(\022)1500 2182 y Fs(p)c Fr(\000)f(j)p Fs(D)s Fr(j)f(\000)i Fu(1)1568 2318 y Fs(m)f Fr(\000)h(j)p Fs(D)s Fr(j)1981 2109 y Fm(\023)2082 2250 y Fu(=)k(\()p Fr(\000)p Fu(1\))2387 2209 y Fn(m)p Fk(\000j)p Fn(D)r Fk(j)2608 2109 y Fm(\022)2681 2182 y Fr(j)p Fs(B)5 b Fr(j)22 b(\000)h(j)p Fs(D)s Fr(j)e(\000)h Fu(1)2791 2318 y Fs(m)h Fr(\000)f(j)p Fs(D)s Fr(j)3247 2109 y Fm(\023)3320 2250 y Fs(:)120 2488 y Fu(Therefore,)334 2726 y Fs(T)391 2741 y Fn(X)459 2726 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)693 2741 y Fn(A)746 2753 y Fh(k)787 2726 y Fu(\))28 b Fr(\000)-16 b(!)1124 2631 y Fm(X)1041 2851 y Fn(D)r Fk(2)p Fl([)p Fn(B)s Fl(])1244 2833 y Fi(\024)p Fh(m)1350 2726 y Fu(\()p Fr(\000)p Fu(1\))1552 2685 y Fn(m)p Fk(\000j)p Fn(D)r Fk(j)1773 2586 y Fm(\022)1846 2659 y Fr(j)p Fs(B)5 b Fr(j)22 b(\000)h(j)p Fs(D)s Fr(j)e(\000)h Fu(1)1956 2794 y Fs(m)h Fr(\000)f(j)p Fs(D)s Fr(j)2412 2586 y Fm(\023)2486 2726 y Fs(f)11 b Fu(\()p Fs(\037)2644 2741 y Fn(D)2707 2726 y Fu(\))28 b(=)f Fs(f)11 b Fu(\()p Fs(\037)3034 2741 y Fn(B)3095 2726 y Fu(\))p Fs(:)3463 3020 y Fg(\003)220 3229 y Fu(It)37 b(is)g(straigh)m(tforw)m(ard)g(that) g Fs(T)1397 3244 y Fn(X)1502 3229 y Fu(is)g(linear)f(and)h(b)s(ounded)h (op)s(erator,)g(so)g(the)g(pro)s(of)e(of)120 3375 y(Theorem)532 3375 y SDict begin H.S end 532 3375 a Black Fu(3.1)p Black 657 3311 a SDict begin H.R end 657 3311 a 657 3375 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.1) cvn H.B /ANN pdfmark end 657 3375 a 32 w Fu(is)c(completed.)220 3520 y(If)g(w)m(e)i(ha)m(v)m(e)g(a)e(subset)i Fs(Y)49 b Fr(\032)28 b Fs(X)8 b Fu(,)33 b(then)g(w)m(e)g(ha)m(v)m(e)h(a)f (natural)e(isometric)g(em)m(b)s(edding)1161 3731 y Fs(e)1206 3690 y Fn(X)1206 3756 y(Y)1301 3731 y Fu(:)d Fs(C)7 b Fu(\()p Fs(\033)1526 3746 y Fn(m)1593 3731 y Fu(\(2)1680 3690 y Fn(Y)1740 3731 y Fu(\)\))28 b Fr(\000)-16 b(!)27 b Fs(C)7 b Fu(\()p Fs(\033)2202 3746 y Fn(m)2269 3731 y Fu(\(2)2356 3690 y Fn(X)2423 3731 y Fu(\)\))120 3943 y(giv)m(en)33 b(b)m(y)g Fs(e)555 3906 y Fn(X)555 3968 y(Y)623 3943 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)857 3958 y Fn(A)913 3943 y Fu(\))28 b(=)g Fs(f)11 b Fu(\()p Fs(\037)1241 3958 y Fn(A)p Fk(\\)p Fn(Y)1401 3943 y Fu(\),)33 b(and)f(also)g(a)g(natural)g(restriction)f(op)s(erator)1160 4154 y Fs(r)1207 4113 y Fn(X)1204 4179 y(Y)1302 4154 y Fu(:)d Fs(C)7 b Fu(\()p Fs(\033)1527 4169 y Fn(m)1594 4154 y Fu(\(2)1681 4113 y Fn(X)1748 4154 y Fu(\)\))27 b Fr(\000)-16 b(!)27 b Fs(C)7 b Fu(\()p Fs(\033)2209 4169 y Fn(m)2276 4154 y Fu(\(2)2363 4113 y Fn(Y)2424 4154 y Fu(\)\))120 4365 y(giv)m(en)30 b(b)m(y)g Fs(r)551 4329 y Fn(X)548 4390 y(Y)619 4365 y Fu(\()p Fs(f)11 b Fu(\)\()p Fs(\037)853 4380 y Fn(A)909 4365 y Fu(\))28 b(=)f Fs(f)11 b Fu(\()p Fs(\037)1236 4380 y Fn(A)1293 4365 y Fu(\).)43 b(These)31 b(t)m(w)m(o)f(op)s(erators)g(ha)m(v)m(e)h (norm)e(one.)43 b(If)29 b(w)m(e)i(ha)m(v)m(e)g(an)120 4510 y(arbitrary)23 b(extension)i(op)s(erator)f Fs(T)41 b Fu(:)28 b Fs(C)7 b Fu(\()p Fs(\033)1657 4525 y Fn(m)1724 4510 y Fu(\(2)1811 4474 y Fn(X)1878 4510 y Fu(\)\))28 b Fr(\000)-17 b(!)28 b Fs(C)7 b Fu(\()p Fs(\033)2340 4525 y Fn(n)2387 4510 y Fu(\(2)2474 4474 y Fn(X)2541 4510 y Fu(\)\),)26 b(then)f(a)f(new)h Fq(r)-5 b(estricte)g(d)120 4656 y Fu(extension)33 b(op)s(erator)1151 4816 y Fs(T)14 b Fr(j)1250 4831 y Fn(Y)1338 4816 y Fu(:)27 b Fs(C)7 b Fu(\()p Fs(\033)1562 4831 y Fn(m)1629 4816 y Fu(\(2)1716 4775 y Fn(Y)1777 4816 y Fu(\)\))27 b Fr(\000)-16 b(!)28 b Fs(C)7 b Fu(\()p Fs(\033)2239 4831 y Fn(n)2286 4816 y Fu(\(2)2373 4775 y Fn(Y)2433 4816 y Fu(\)\))120 5002 y(is)34 b(induced)g(as)g Fs(T)14 b Fr(j)804 5017 y Fn(Y)894 5002 y Fu(=)30 b Fs(r)1047 4966 y Fn(X)1044 5027 y(Y)1138 5002 y Fr(\016)23 b Fs(S)28 b Fr(\016)23 b Fs(e)1417 4966 y Fn(X)1417 5027 y(Y)1485 5002 y Fu(.)48 b(This)34 b(new)h(extension)g(op)s(erator)e(satis\014es)i Fr(k)p Fs(T)14 b Fr(j)3323 5017 y Fn(Y)3383 5002 y Fr(k)30 b(\024)120 5147 y(k)p Fs(T)14 b Fr(k)p Fu(,)32 b(so)h(in)f(particular)e(w)m(e)k (ha)m(v)m(e:)p Black Black eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 232 -8 a SDict begin H.S end 232 -8 a 232 -8 a SDict begin H.R end 232 -8 a 232 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end 232 -8 a Black 240 70 a Fo(12)786 b(ANTONIO)24 b(A)-9 b(VIL)1693 52 y(\023)1686 70 y(ES,)25 b(WITOLD)g(MAR)n(CISZEWSKI)p Black 240 266 a SDict begin H.S end 240 266 a 240 266 a SDict begin 17.5 H.A end 240 266 a 240 266 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.6) cvn /DEST pdfmark end 240 266 a Black FA(Prop)s(osition)38 b(3.6.)p Black 41 w Fq(If)d Fr(j)p Fs(Y)21 b Fr(j)27 b(\024)h(j)p Fs(X)8 b Fr(j)p Fq(,)34 b(then)h Fs(\021)t Fu(\()p Fs(\033)2001 281 y Fn(m)2067 266 y Fu(\(2)2154 230 y Fn(Y)2215 266 y Fu(\))p Fs(;)17 b(\033)2352 281 y Fn(n)2399 266 y Fu(\(2)2486 230 y Fn(Y)2546 266 y Fu(\)\))28 b Fr(\024)g Fs(\021)t Fu(\()p Fs(\033)2900 281 y Fn(m)2967 266 y Fu(\(2)3054 230 y Fn(X)3121 266 y Fu(\))p Fs(;)17 b(\033)3258 281 y Fn(n)3305 266 y Fu(\(2)3392 230 y Fn(X)3459 266 y Fu(\)\))p Fq(.)240 502 y SDict begin H.S end 240 502 a 240 502 a SDict begin 17.5 H.A end 240 502 a 240 502 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.7) cvn /DEST pdfmark end 240 502 a Black FA(Theorem)37 b(3.7.)p Black 41 w Fq(Fix)c Fs(m)28 b(<)g(n)g Fr(\024)g Fs(p)f(<)h(!)37 b Fq(and)c Fs(")28 b(>)f Fu(0)p Fq(.)45 b(If)33 b Fs(T)41 b Fu(:)28 b Fs(C)7 b Fu(\()p Fs(\033)2700 517 y Fn(m)2767 502 y Fu(\(2)2854 465 y Fn(X)2921 502 y Fu(\)\))28 b Fr(\000)-17 b(!)28 b Fs(C)7 b Fu(\()p Fs(\033)3383 517 y Fn(n)3430 502 y Fu(\(2)3517 465 y Fn(X)3584 502 y Fu(\)\))240 647 y Fq(is)38 b(any)g(extension)f(op)-5 b(er)g(ator)38 b(and)g Fr(j)p Fs(X)8 b Fr(j)33 b(\025)h(@)1902 662 y Fn(p)p Fk(\000)p Fl(1)2032 647 y Fq(,)39 b(then)f(ther)-5 b(e)38 b(exists)g Fs(Y)55 b Fr(2)35 b Fu([)p Fs(X)8 b Fu(])3193 611 y Fn(p)3271 647 y Fq(such)38 b(that)240 792 y Fr(k)p Fs(T)14 b Fr(j)389 807 y Fn(Y)460 792 y Fr(\000)d Fs(T)605 807 y Fn(Y)666 792 y Fr(k)27 b Fs(<)h(")p Fq(,)i(wher)-5 b(e)29 b Fs(T)14 b Fr(j)1322 807 y Fn(Y)1412 792 y Fq(is)29 b(the)h(r)-5 b(estricte)g(d)30 b(op)-5 b(er)g(ator,)30 b(and)f Fs(T)2739 807 y Fn(Y)2830 792 y Fq(is)g(the)h(op)-5 b(er)g(ator)29 b(fr)-5 b(om)240 937 y(The)g(or)g(em)645 937 y SDict begin H.S end 645 937 a Black Fq(3.1)p Black 774 873 a SDict begin H.R end 774 873 a 774 937 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.1) cvn H.B /ANN pdfmark end 774 937 a Fq(.)p Black 240 1173 a(Pr)g(o)g(of.)p Black 41 w Fu(W)d(e)33 b(\014x)g Fs(")28 b(>)f Fu(0)32 b(and)h(w)m(e)h(tak)m(e)f(a)f(m)m(uc)m(h)i(smaller)1692 1401 y Fs(")1738 1360 y Fk(0)1789 1401 y Fs(<)2014 1334 y(")p 1902 1378 269 4 v 1902 1470 a Fu(2)1951 1441 y Fl(\()p Fn(p)p Fl(+2\))2131 1422 y Fj(2)2181 1401 y Fs(:)240 1642 y Fu(Consider)d(the)g(measure-v)-5 b(alued)29 b(function)h Fs(')1938 1657 y Fn(T)2021 1642 y Fu(:)e Fs(\033)2131 1657 y Fn(n)2178 1642 y Fu(\(2)2265 1606 y Fn(X)2332 1642 y Fu(\))g Fr(\000)-16 b(!)27 b Fs(M)10 b Fu(\()p Fs(\033)2783 1657 y Fn(m)2851 1642 y Fu(\(2)2938 1606 y Fn(X)3005 1642 y Fu(\)\))30 b(asso)s(ciated)g(to)240 1787 y(the)h(op)s(erator)g Fs(T)14 b Fu(,)31 b(giv)m(en)g(b)m(y)h Fs(')1378 1802 y Fn(T)1433 1787 y Fu(\()p Fs(\037)1532 1802 y Fn(A)1589 1787 y Fu(\))c(=)f Fs(T)1829 1751 y Fk(\003)1868 1787 y Fu(\()p Fs(\016)1949 1802 y Fn(\037)1993 1813 y Fh(A)2048 1787 y Fu(\).)43 b(F)-8 b(or)30 b(ev)m(ery)j Fs(A)28 b Fr(2)g Fu([)p Fs(X)8 b Fu(])2923 1751 y Fk(\024)p Fn(n)3056 1787 y Fu(c)m(ho)s(ose)31 b(a)g(\014nite)240 1933 y(set)i Fs(S)458 1897 y Fl(1)452 1958 y Fn(A)537 1933 y Fr(\032)28 b Fs(X)i Fr(n)22 b Fs(A)33 b Fu(suc)m(h)h(that)1041 2160 y Fr(j)p Fs(')1133 2175 y Fn(T)1188 2160 y Fu(\()p Fs(\037)1287 2175 y Fn(A)1344 2160 y Fu(\))p Fr(jf)p Fs(\037)1521 2175 y Fn(C)1607 2160 y Fr(2)29 b Fs(\033)1757 2175 y Fn(m)1824 2160 y Fu(\(2)1911 2119 y Fn(X)1978 2160 y Fu(\))e(:)h Fs(C)35 b Fr(6\032)28 b Fs(A)22 b Fr([)h Fs(S)2558 2119 y Fl(1)2552 2185 y Fn(A)2609 2160 y Fr(g)k Fs(<)h(")2836 2119 y Fk(0)340 2388 y Fu(F)-8 b(or)47 b Fs(")54 b(>)g Fu(0,)d(and)e(reals)e Fs(a;)17 b(b)p Fu(,)53 b(w)m(e)c(will)d(write)i Fs(a)2303 2332 y Fn(")2281 2388 y Fr(\030)55 b Fs(b)48 b Fu(if)f Fr(j)p Fs(a)33 b Fr(\000)g Fs(b)p Fr(j)54 b Fs(<)g(")p Fu(.)90 b(F)-8 b(or)47 b(ev)m(ery)240 2533 y Fs(A;)17 b(B)33 b Fr(2)28 b Fu([)p Fs(X)8 b Fu(])701 2497 y Fk(\024)p Fn(m)855 2533 y Fu(de\014ne)931 2761 y SDict begin H.S end 931 2761 a 931 2761 a SDict begin 17.5 H.A end 931 2761 a 931 2761 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.13) cvn /DEST pdfmark end 931 2761 a Fs(O)1006 2776 y Fn(B)1094 2761 y Fu(=)27 b Fr(f)p Fs(\037)1308 2776 y Fn(C)1395 2761 y Fr(2)h Fs(\033)1544 2776 y Fn(m)1611 2761 y Fu(\(2)1698 2720 y Fn(X)1765 2761 y Fu(\))11 b(:)33 b Fs(B)g Fr(\022)28 b Fs(C)7 b Fr(g)17 b Fs(;)p Black -2017 w Fu(\(3.13\))p Black 752 2948 a Fs(U)818 2963 y Fn(A)903 2948 y Fu(=)27 b Fr(f)p Fs(\026)g Fr(2)i Fs(M)10 b Fu(\()p Fs(\033)1434 2963 y Fn(m)1501 2948 y Fu(\(2)1588 2907 y Fn(X)1655 2948 y Fu(\)\))h(:)34 b Fs(\026)p Fu(\()p Fs(O)1975 2963 y Fn(B)2034 2948 y Fu(\))2111 2892 y Fn(")2144 2868 y Fi(0)2100 2948 y Fr(\030)28 b Fu(1)65 b(for)32 b(all)f Fs(B)h Fr(\022)d Fs(A)p Fr(g)17 b Fs(:)240 3176 y Fu(Since)31 b(ev)m(ery)i Fs(O)824 3191 y Fn(B)916 3176 y Fu(suc)m(h)f(that)f Fs(B)i Fr(\022)28 b Fs(A)p Fu(,)k(is)f(a)g(clop)s (en)f(neigh)m(b)s(orho)s(o)s(d)h(of)f Fs(\037)2951 3191 y Fn(A)3040 3176 y Fu(in)g Fs(\033)3207 3191 y Fn(m)3274 3176 y Fu(\(2)3361 3140 y Fn(X)3428 3176 y Fu(\),)i(the)240 3321 y(set)41 b Fs(U)466 3336 y Fn(A)564 3321 y Fu(is)g(an)f(op)s(en)h (neigh)m(b)s(orho)s(o)s(d)f(of)g Fs(\016)1841 3336 y Fn(\037)1885 3347 y Fh(A)1980 3321 y Fu(in)g Fs(M)10 b Fu(\()p Fs(\033)2299 3336 y Fn(m)2367 3321 y Fu(\(2)2454 3285 y Fn(X)2521 3321 y Fu(\)\).)68 b(De\014ne)42 b Fs(V)3060 3336 y Fn(A)3158 3321 y Fu(=)g Fs(')3340 3280 y Fk(\000)p Fl(1)3340 3348 y Fn(T)3434 3321 y Fu(\()p Fs(U)3538 3336 y Fn(A)3595 3321 y Fu(\).)240 3466 y(Since)33 b Fs(\037)556 3481 y Fn(A)641 3466 y Fr(2)28 b Fs(V)792 3481 y Fn(A)848 3466 y Fu(,)33 b(w)m(e)h(can)f(\014nd)g(a)f(\014nite)g(set)i Fs(S)1975 3430 y Fl(2)1969 3492 y Fn(A)2053 3466 y Fr(\022)28 b Fs(X)i Fr(n)22 b Fs(A)33 b Fu(suc)m(h)h(that)738 3694 y SDict begin H.S end 738 3694 a 738 3694 a SDict begin 17.5 H.A end 738 3694 a 738 3694 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.14) cvn /DEST pdfmark end 738 3694 a Fs(W)830 3709 y Fn(A)915 3694 y Fu(=)27 b Fr(f)p Fs(\037)1129 3709 y Fn(B)1217 3694 y Fr(2)i Fs(\033)1367 3709 y Fn(n)1414 3694 y Fu(\(2)1501 3653 y Fn(X)1568 3694 y Fu(\))11 b(:)33 b Fs(A)28 b Fr(\022)g Fs(B)38 b Fu(and)33 b Fs(B)27 b Fr(\\)22 b Fs(S)2440 3653 y Fl(2)2434 3719 y Fn(A)2519 3694 y Fu(=)28 b Fr(;g)f(\022)h Fs(V)2912 3709 y Fn(A)2969 3694 y Fs(:)p Black -2756 w Fu(\(3.14\))p Black 340 3922 a(F)-8 b(or)31 b(ev)m(ery)k Fs(A)28 b Fr(2)g Fu([)p Fs(X)8 b Fu(])1110 3886 y Fk(\024)p Fn(n)1234 3922 y Fr(n)22 b Fu([)p Fs(X)8 b Fu(])1449 3886 y Fk(\024)p Fn(m)1570 3922 y Fu(,)33 b(w)m(e)g(put)g Fs(S)2018 3886 y Fl(2)2012 3947 y Fn(A)2097 3922 y Fu(=)27 b Fr(;)p Fu(.)340 4067 y(Let)g Fs(S)569 4082 y Fn(A)654 4067 y Fu(=)g Fs(S)823 4031 y Fl(1)817 4092 y Fn(A)886 4067 y Fr([)11 b Fs(S)1029 4031 y Fl(2)1023 4092 y Fn(A)1081 4067 y Fu(,)28 b(for)f Fs(A)h Fr(2)g Fu([)p Fs(X)8 b Fu(])1618 4031 y Fk(\024)p Fn(n)1720 4067 y Fu(.)41 b(Using)27 b(Lemma)2400 4067 y SDict begin H.S end 2400 4067 a Black Fu(2.2)p Black 2525 4003 a SDict begin H.R end 2525 4003 a 2525 4067 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.2) cvn H.B /ANN pdfmark end 2525 4067 a 27 w Fu(w)m(e)i(can)e(\014nd)h(a)f(set)h Fs(Y)49 b Fr(2)28 b Fu([)p Fs(X)8 b Fu(])3620 4031 y Fn(p)240 4212 y Fu(suc)m(h)34 b(that)e Fs(Y)44 b Fr(\\)22 b Fs(S)920 4227 y Fn(A)1005 4212 y Fu(=)28 b Fr(;)k Fu(for)g(all)e Fs(A)e Fr(2)g Fu([)p Fs(Y)22 b Fu(])1803 4176 y Fk(\024)p Fn(m)1924 4212 y Fu(.)340 4358 y(It)33 b(is)f(enough)h(to)g(pro)m(v)m (e)h(that)e Fr(k)p Fs(')1588 4373 y Fn(T)10 b Fk(j)1659 4384 y Fh(Y)1716 4358 y Fu(\()p Fs(\037)1815 4373 y Fn(Z)1872 4358 y Fu(\))22 b Fr(\000)h Fs(')2096 4373 y Fn(T)2137 4384 y Fh(Y)2194 4358 y Fu(\()p Fs(\037)2293 4373 y Fn(Z)2350 4358 y Fu(\))p Fr(k)28 b Fs(<)g(")k Fu(for)g(ev)m(ery)j Fs(Z)g Fr(2)28 b Fu([)p Fs(Y)22 b Fu(])3384 4322 y Fk(\024)p Fn(n)3486 4358 y Fu(.)44 b(So)240 4503 y(let)36 b(us)i(\014x)g Fs(Z)k Fr(2)35 b Fu([)p Fs(Y)22 b Fu(])1001 4467 y Fk(\024)p Fn(n)1103 4503 y Fu(.)56 b(If)37 b Fr(j)p Fs(Z)7 b Fr(j)35 b(\024)g Fs(m)j Fu(then)f(w)m(e)h(ha)m(v)m(e)g Fs(')2355 4518 y Fn(T)10 b Fk(j)2426 4529 y Fh(Y)2483 4503 y Fu(\()p Fs(\037)2582 4518 y Fn(Z)2639 4503 y Fu(\))35 b(=)g Fs(')2887 4518 y Fn(T)2928 4529 y Fh(Y)2986 4503 y Fu(\()p Fs(\037)3085 4518 y Fn(Z)3141 4503 y Fu(\))g(=)g Fs(\016)3368 4518 y Fn(\037)3412 4529 y Fh(Z)3503 4503 y Fu(and)240 4648 y(w)m(e)f(are)g(done.)45 b(So)34 b(w)m(e)g(supp)s(ose)g(that)f Fs(m)d(<)e Fr(j)p Fs(Z)7 b Fr(j)28 b(\024)i Fs(n)p Fu(.)45 b(Then,)35 b(b)m(y)f(the)g(de\014nition)e(of)h Fs(T)3485 4663 y Fn(Y)3579 4648 y Fu(in)240 4794 y(Theorem)652 4794 y SDict begin H.S end 652 4794 a Black Fu(3.1)p Black 777 4729 a SDict begin H.R end 777 4729 a 777 4794 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.1) cvn H.B /ANN pdfmark end 777 4794 a 922 5029 a Fs(')986 5044 y Fn(T)1027 5055 y Fh(Y)1085 5029 y Fu(\()p Fs(\037)1184 5044 y Fn(Z)1241 5029 y Fu(\))27 b(=)1489 4934 y Fm(X)1410 5154 y Fn(B)s Fk(2)p Fl([)p Fn(Z)5 b Fl(])1606 5135 y Fi(\024)p Fh(m)1712 5029 y Fu(\()p Fr(\000)p Fu(1\))1914 4988 y Fn(m)p Fk(\000j)p Fn(B)s Fk(j)2131 4888 y Fm(\022)2204 4961 y Fr(j)p Fs(Z)i Fr(j)22 b(\000)g(j)p Fs(B)5 b Fr(j)22 b(\000)h Fu(1)2311 5097 y Fs(m)g Fr(\000)g(j)p Fs(B)5 b Fr(j)2760 4888 y Fm(\023)2834 5029 y Fs(\016)2877 5044 y Fn(\037)2921 5055 y Fh(B)p Black Black eop end end %%Page: 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Fu(1\))1386 3991 y Fn(m)p Fk(\000j)p Fn(C)5 b Fk(j)1602 3892 y Fm(\022)1675 3965 y Fr(j)p Fs(Z)i Fr(j)22 b(\000)g(j)p Fs(C)7 b Fr(j)22 b(\000)h Fu(1)1783 4101 y Fs(m)f Fr(\000)h(j)p Fs(C)7 b Fr(j)2229 3892 y Fm(\023)2400 4032 y Fu(when)34 b Fr(j)p Fs(C)7 b Fr(j)27 b Fs(>)g Fr(j)p Fs(B)5 b Fr(j)p Fs(:)120 4291 y Fu(On)31 b(the)g(other)h(hand,)f(b)m(y)h(equation)f(\()1539 4291 y SDict begin H.S end 1539 4291 a Black Fu(3.4)p Black 1663 4227 a SDict begin H.R end 1663 4227 a 1663 4291 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 1663 4291 a Fu(\))g(in)g(Lemma)2191 4291 y SDict begin H.S end 2191 4291 a Black Fu(3.3)p Black 2316 4227 a SDict begin H.R end 2316 4227 a 2316 4291 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.3) cvn H.B /ANN pdfmark end 2316 4291 a Fu(,)g(applied)f(to)h Fs(k)f Fu(=)e Fs(m)19 b Fr(\000)g(j)p Fs(B)5 b Fr(j)31 b Fu(and)120 4437 y Fs(l)f Fu(=)d Fr(j)p Fs(Z)7 b Fr(j)22 b(\000)h Fs(m)p Fu(,)32 b(and)h Fs(i)g Fu(running)f(o)m(v)m(er)i(all)c (p)s(ossible)i(cardinalities)e(of)i Fs(C)d Fr(n)22 b Fs(B)5 b Fu(,)503 4714 y(\()p Fs(?)22 b(?)g(?)p Fu(\))64 b(1)83 b(=)1166 4590 y Fn(m)1125 4620 y Fm(X)1140 4830 y Fn(i)p Fl(=0)1269 4714 y Fu(\()p Fr(\000)p Fu(1\))1471 4673 y Fn(m)p Fk(\000j)p Fn(B)s Fk(j\000)p Fn(i)1767 4574 y Fm(\022)1841 4647 y Fr(j)p Fs(Z)7 b Fr(j)21 b(\000)i(j)p Fs(B)5 b Fr(j)2017 4783 y Fs(i)2226 4574 y Fm(\023\022)2373 4647 y Fr(j)p Fs(Z)i Fr(j)21 b(\000)i(j)p Fs(B)5 b Fr(j)21 b(\000)i Fs(i)f Fr(\000)h Fu(1)2480 4783 y Fs(m)f Fr(\000)h(j)p Fs(B)5 b Fr(j)22 b(\000)g Fs(i)3084 4574 y Fm(\023)966 5029 y Fu(=)1259 4934 y Fm(X)1125 5154 y Fn(B)s Fk(\032)p Fn(C)5 b Fk(2)p Fl([)p Fn(Z)g Fl(])1431 5135 y Fi(\024)p Fh(m)1537 5029 y Fu(\()p Fr(\000)p Fu(1\))1739 4988 y Fn(m)p Fk(\000j)p Fn(C)g Fk(j)1955 4888 y Fm(\022)2028 4961 y Fr(j)p Fs(Z)i Fr(j)21 b(\000)i(j)p Fs(C)7 b Fr(j)21 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/Link/Dest (thm.3.10) cvn H.B /ANN pdfmark end 3540 2700 a 120 2845 a Fu(comm)m(utes)39 b(for)g(b)s(oth)f(the)i (op)s(erators)1609 2820 y(~)1590 2845 y Fs(T)1647 2860 y Fk(\003)1725 2845 y Fu(and)g Fs(T)1979 2860 y Fk(\003)2018 2845 y Fu(,)h(so)e(the)h(inequalit)m(y)e(is)g(transferred)i(to)120 2991 y Fr(k)189 2966 y Fu(~)170 2991 y Fs(T)227 3006 y Fn(X)317 2991 y Fr(\000)22 b Fs(T)473 3006 y Fn(X)541 2991 y Fr(k)27 b Fs(<)h(")p Fu(.)2668 b Fg(\003)120 3147 y SDict begin H.S end 120 3147 a 120 3147 a SDict begin 17.5 H.A end 120 3147 a 120 3147 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 120 3147 a 719 3267 a Fu(4.)49 b Ft(Extension)37 b(opera)-7 b(tors)37 b(on)h(sp)-7 b(a)n(ces)37 b Fs(C)7 b Fu(\()p Fs(\033)2630 3282 y Fn(m)2696 3267 y Fu(\(2)2783 3231 y Fk(@)2826 3240 y Fj(1)2865 3267 y Fu(\)\))120 3471 y SDict begin H.S end 120 3471 a 120 3471 a SDict begin 17.5 H.A end 120 3471 a 120 3471 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.1) cvn /DEST pdfmark end 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b(section)g(is)g(dev)m(oted)h(to)f(the)h(pro)s(of)e(of)g (Theorem)2554 3181 y SDict begin H.S end 2554 3181 a Black Fu(1.2)p Black 2678 3117 a SDict begin H.R end 2678 3117 a 2678 3181 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.1.2) cvn H.B /ANN pdfmark end 2678 3181 a Fu(.)91 b(First,)51 b(w)m(e)e(can)f(get)g(a)240 3326 y(simpler)32 b(top)s(ologically)d(equiv)-5 b(alen)m(t)34 b(description)f(of)g(the)g (balls)g(in)f(Hilb)s(ert)g(space.)47 b(W)-8 b(e)34 b(\014x)240 3471 y(an)39 b(uncoun)m(table)h(set)g Fs(X)8 b Fu(.)63 b(F)-8 b(or)39 b Fs(\025)g Fr(2)g Fu(\(0)p Fs(;)17 b Fu(+)p Fr(1)p Fu(\),)40 b(let)e Fs(B)2293 3486 y Fn(\025)2378 3471 y Fu(=)h Fr(f)p Fs(z)44 b Fr(2)39 b Fe(R)2803 3435 y Fn(X)2915 3471 y Fu(:)2982 3397 y Fm(P)3087 3500 y Fn(i)p Fk(2)p Fn(X)3242 3471 y Fr(j)p Fs(z)3315 3486 y Fn(i)3343 3471 y Fr(j)g(\024)g Fs(\025)p Fr(g)p Fu(.)240 3617 y(If)j(\001)h(:)h Fs(`)583 3632 y Fl(2)622 3617 y Fu(\()p Fs(X)8 b Fu(\))43 b Fr(\000)-16 b(!)43 b Fe(R)1100 3580 y Fn(X)1215 3617 y Fu(is)f(the)g(function)f(giv)m(en)h(b)m(y)h (\001\(\()p Fs(z)2502 3632 y Fn(i)2531 3617 y Fu(\))2569 3632 y Fn(i)p Fk(2)p Fn(X)2707 3617 y Fu(\))g(=)h(\()p Fs(sig)t(n)p Fu(\()p Fs(z)3217 3632 y Fn(i)3245 3617 y Fu(\))28 b Fr(\001)g Fs(z)3416 3580 y Fl(2)3412 3641 y Fn(i)3457 3617 y Fu(\))3495 3632 y Fn(i)p Fk(2)p Fn(X)3633 3617 y Fu(,)240 3762 y(then)35 b(it)d(is)i(easy)h(to)e(c)m(hec)m(k)j (that)e(\001)c(:)g Fs(\025B)1774 3777 y Fn(`)1803 3786 y Fj(2)1838 3777 y Fl(\()p Fn(X)5 b Fl(\))1990 3762 y Fr(\000)-16 b(!)29 b Fs(B)2254 3734 y Fk(p)p 2313 3734 42 3 v 56 x Fn(\025)2392 3762 y Fu(establishes)35 b(a)e(homeomorphism) 240 3907 y(for)h(eac)m(h)h Fs(\025)p Fu(.)49 b(Th)m(us,)37 b(Theorem)1435 3907 y SDict begin H.S end 1435 3907 a Black Fu(1.2)p Black 1560 3843 a SDict begin H.R end 1560 3843 a 1560 3907 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.1.2) cvn H.B /ANN pdfmark end 1560 3907 a 34 w Fu(can)e(b)s(e)g(equiv)-5 b(alen)m(tly)34 b(reform)m(ulated)f(sa)m (ying)h(that)g(for)240 4053 y(eac)m(h)40 b(0)f Fs(<)f(\025)h(<)g(\026)f (<)h Fu(+)p Fr(1)p Fu(,)h(there)g(is)e(no)h(extension)h(op)s(erator)f Fs(T)52 b Fu(:)39 b Fs(C)7 b Fu(\()p Fs(B)3038 4068 y Fn(\025)3083 4053 y Fu(\))39 b Fr(\000)-16 b(!)38 b Fs(C)7 b Fu(\()p Fs(B)3548 4068 y Fn(\026)3595 4053 y Fu(\).)240 4198 y(W)-8 b(e)38 b(can)g(also)f(lo)s(ok)f(at)i(the)g(compact)f(sets)i Fs(B)1982 4157 y Fl(+)1977 4226 y Fn(\025)2078 4198 y Fu(=)d Fr(f)p Fs(z)41 b Fr(2)c Fs(B)2503 4213 y Fn(\025)2585 4198 y Fu(:)f Fr(8)p Fs(i)h Fr(2)g Fs(X)45 b(z)3047 4213 y Fn(i)3112 4198 y Fr(\025)37 b Fu(0)p Fr(g)p Fu(.)59 b(There)240 4343 y(is)37 b(a)h(con)m(tin)m(uous)h(retraction)e Fs(\032)g Fu(:)g Fs(B)1604 4358 y Fn(\025)1686 4343 y Fr(\000)-16 b(!)37 b Fs(B)1963 4302 y Fl(+)1958 4371 y Fn(\025)2060 4343 y Fu(giv)m(en)h(b)m(y)h Fs(\032)p Fu(\(\()p Fs(z)2632 4358 y Fn(i)2660 4343 y Fu(\))2698 4358 y Fn(i)p Fk(2)p Fn(X)2837 4343 y Fu(\))e(=)f(\()p Fr(j)p Fs(z)3135 4358 y Fn(i)3163 4343 y Fr(j)p Fu(\))3229 4358 y Fn(i)p Fk(2)p Fn(X)3368 4343 y Fu(.)59 b(If)38 b(an)240 4488 y(extension)29 b(op)s(erator)f Fs(T)42 b Fu(:)28 b Fs(C)7 b Fu(\()p Fs(B)1399 4503 y Fn(\025)1444 4488 y Fu(\))27 b Fr(\000)-16 b(!)28 b Fs(C)7 b Fu(\()p Fs(B)1887 4503 y Fn(\026)1933 4488 y Fu(\))29 b(exists,)h(then)f Fs(T)2582 4452 y Fl(+)2641 4488 y Fu(\()p Fs(f)11 b Fu(\))27 b(=)h Fs(T)14 b Fu(\()p Fs(f)24 b Fr(\016)14 b Fs(\032)p Fu(\))p Fr(j)3268 4516 y Fn(B)3324 4486 y Fj(+)3321 4526 y Fh(\026)3408 4488 y Fu(w)m(ould)240 4634 y(giv)m(e)34 b(an)f(extension)i(op)s(erator)e Fs(T)1476 4598 y Fl(+)1565 4634 y Fu(:)d Fs(C)7 b Fu(\()p Fs(B)1816 4592 y Fl(+)1811 4661 y Fn(\025)1875 4634 y Fu(\))30 b Fr(\000)-17 b(!)30 b Fs(C)7 b Fu(\()p Fs(B)2327 4598 y Fl(+)2322 4658 y Fn(\026)2386 4634 y Fu(\).)47 b(Th)m(us,)36 b(it)d(is)g(enough)h(to)g (pro)m(v)m(e)240 4779 y(the)f(follo)m(wing:)240 4998 y SDict begin H.S end 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(these)i(that)e(is)120 847 y(relev)-5 b(an)m(t)33 b(to)f(us)h(is)f(the) h(v)-5 b(alue)32 b(of)g(the)h(norm.)120 1069 y SDict begin H.S end 120 1069 a 120 1069 a SDict begin 17.5 H.A end 120 1069 a 120 1069 a SDict begin [/View [/XYZ H.V]/Dest (thm.5.2) cvn /DEST pdfmark end 120 1069 a Black FA(Lemma)39 b(5.2.)p Black 42 w Fq(F)-7 b(or)34 b(e)-5 b(ach)34 b Fs(m)28 b(<)g(!)t Fq(,)34 b(ther)-5 b(e)35 b(exists)f(a)h(r)-5 b(e)g(gular)35 b(extension)e(op)-5 b(er)g(ator)1345 1272 y Fs(R)28 b Fu(:)g Fs(C)7 b Fu(\()p Fs(S)1677 1287 y Fn(m)1744 1272 y Fu(\))27 b Fr(\000)-16 b(!)27 b Fs(C)7 b Fu(\()p Fs(B)2191 1231 y Fl(+)2186 1297 y(1)2250 1272 y Fu(\))p Fs(:)p Black 120 1494 a Fq(Pr)-5 b(o)g(of.)p Black 41 w Fu(Fix)27 b Fs(")g Fu(suc)m(h)i(that)e(1)p Fs(=)p Fu(\()p Fs(m)12 b Fu(+)g(1\))27 b Fs(<)h(")f(<)h Fu(1)p Fs(=m)p Fu(.)41 b(Cho)s(ose)29 b(a)e(con)m(tin)m(uous)h (nondecreasing)120 1639 y(function)37 b Fs(g)554 1654 y Fl(1)629 1639 y Fu(:)f Fe(R)47 b Fr(\000)-17 b(!)36 b Fu([0)p Fs(;)17 b Fu(1])37 b(suc)m(h)h(that)g Fs(g)1717 1654 y Fl(1)1756 1639 y Fu(\()p Fs(t)p Fu(\))e(=)f(0)i(if)g Fs(t)f Fr(\024)g Fs(")h Fu(and)g Fs(g)t Fu(\()p Fs(t)p Fu(\))f(=)f(1)i(if)g Fs(t)f Fr(\025)g Fu(1)p Fs(=m)p Fu(.)120 1785 y(Let)d(also)e Fs(g)537 1800 y Fl(0)604 1785 y Fu(=)d(1)22 b Fr(\000)g Fs(g)925 1800 y Fl(1)964 1785 y Fu(.)44 b(F)-8 b(or)32 b Fs(z)g Fu(=)c(\()p Fs(z)1474 1800 y Fn(i)1502 1785 y Fu(\))1540 1800 y Fn(i)p Fk(2)p Fn(X)1706 1785 y Fr(2)g Fs(B)1874 1800 y Fl(1)1914 1785 y Fu(,)k(de\014ne)i(the)f(set)1168 1988 y Fs(F)1231 2003 y Fn(z)1299 1988 y Fu(=)27 b Fr(f)p Fs(i)h Fr(2)g Fs(X)36 b Fu(:)27 b Fs(z)1823 2003 y Fn(i)1880 1988 y Fs(>)g(")p Fr(g)g(2)h Fu([)p Fs(X)8 b Fu(])2343 1947 y Fk(\024)p Fn(m)2465 1988 y Fs(:)120 2191 y Fu(The)34 b(op)s(erator)d Fs(R)j Fu(is)e(giv)m(en)h(b)m(y)g(the)g(follo)m(wing)d(form)m(ula:)931 2527 y Fs(R)q(f)11 b Fu(\()p Fs(z)t Fu(\))28 b(=)1343 2432 y Fm(X)1321 2643 y Fn(A)p Fk(\032)p Fn(F)1474 2651 y Fh(z)1526 2356 y Fm( )1617 2432 y(Y)1605 2643 y Fn(i)p Fk(2)p Fn(F)1721 2651 y Fh(z)1773 2527 y Fs(g)1820 2542 y Fn(\037)1864 2553 y Fh(A)1914 2542 y Fl(\()p Fn(i)p Fl(\))1997 2527 y Fu(\()p Fs(z)2080 2542 y Fn(i)2108 2527 y Fu(\))2146 2356 y Fm(!)2247 2527 y Fr(\001)22 b Fs(f)11 b Fu(\()p Fs(m)2479 2485 y Fk(\000)p Fl(1)2573 2527 y Fs(\037)2634 2542 y Fn(A)2691 2527 y Fu(\))120 2777 y(\(here,)33 b(w)m(e)h(use)g(the)f(con)m(v)m(en)m(tion)g(that)1576 2702 y Fm(Q)1670 2806 y Fn(i)p Fk(2)p Fn(F)1786 2814 y Fh(z)1842 2777 y Fs(g)1889 2793 y Fn(\037)1933 2804 y Fh(A)1983 2793 y Fl(\()p Fn(i)p Fl(\))2066 2777 y Fu(\()p Fs(z)2149 2792 y Fn(i)2177 2777 y Fu(\))28 b(=)f(1,)33 b(if)e Fs(F)2607 2792 y Fn(z)2675 2777 y Fu(=)c Fr(;)p Fu(\).)220 2922 y(First,)40 b(w)m(e)g(c)m(hec)m(k)i(that)d Fs(R)q(f)50 b Fu(is)39 b(an)h(extension)g(of)f Fs(f)11 b Fu(.)64 b(Namely)-8 b(,)40 b(if)e Fs(z)44 b Fu(=)39 b Fs(m)3028 2886 y Fk(\000)p Fl(1)3123 2922 y Fs(\037)3184 2937 y Fn(A)3281 2922 y Fr(2)g Fs(S)3446 2937 y Fn(m)3513 2922 y Fu(,)120 3068 y(then)f Fs(F)410 3083 y Fn(z)485 3068 y Fu(=)e Fs(A)h Fu(and)g Fs(g)948 3083 y Fn(\037)992 3094 y Fh(A)1046 3068 y Fu(\()p Fs(i)p Fu(\)\()p Fs(z)1238 3083 y Fn(i)1267 3068 y Fu(\))g(equals)g(0)g(when)i Fs(i)c Fr(62)h Fs(A)i Fu(and)f(equals)h(1)e(when)j Fs(i)d Fr(2)g Fs(A)p Fu(.)57 b(So)120 3213 y Fs(R)q(f)11 b Fu(\()p Fs(z)t Fu(\))38 b(=)g Fs(f)11 b Fu(\()p Fs(z)t Fu(\).)62 b(Second,)42 b(for)c(an)m(y)h Fs(z)k Fr(2)38 b Fs(B)1786 3172 y Fl(+)1781 3237 y(1)1845 3213 y Fu(,)j(since)e Fs(g)2205 3228 y Fl(0)2244 3213 y Fu(\()p Fs(z)2327 3228 y Fn(i)2355 3213 y Fu(\))26 b(+)g Fs(g)2568 3228 y Fl(1)2608 3213 y Fu(\()p Fs(z)2691 3228 y Fn(i)2719 3213 y Fu(\))38 b(=)g(1,)i(it)d(is)i(easy)g(to)120 3358 y(c)m(hec)m(k)c(that)1255 3504 y Fm(X)1233 3716 y Fn(A)p Fk(\032)p Fn(F)1386 3724 y Fh(z)1438 3429 y Fm( )1529 3504 y(Y)1516 3716 y Fn(i)p Fk(2)p Fn(F)1632 3724 y Fh(z)1685 3599 y Fs(g)1732 3615 y Fn(\037)1776 3626 y Fh(A)1825 3615 y Fl(\()p Fn(i)p Fl(\))1908 3599 y Fu(\()p Fs(z)1991 3614 y 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