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href="/nlab/show/functional+analysis">functional analysis</a>, <a class="existingWikiWord" href="/nlab/show/topology">topology</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic analysis</a></p> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+analysis">infinitesimal analysis</a></p> <p><a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a></p> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">Introduction</a></em></p> <h2 id="basic_concepts">Basic concepts</h2> <p><a class="existingWikiWord" href="/nlab/show/triangle+inequality">triangle inequality</a></p> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>, <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> <p><a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a></p> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a>, <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit of a sequence</a></p> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compactness</a>, <a class="existingWikiWord" href="/nlab/show/sequentially+compact+space">sequential compactness</a></p> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/integration">integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <h2 id="basic_facts">Basic facts</h2> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> <p>…</p> <h2 id="theorems">Theorems</h2> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> <p>…</p> </div></div> <h4 id="functional_analysis">Functional analysis</h4> <div class="hide"><div> <ul> <li><strong><a class="existingWikiWord" href="/nlab/show/functional+analysis">Functional Analysis</a></strong></li> </ul> <h2 id="overview_diagrams">Overview diagrams</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TVS+relationships">topological vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diagram+of+LCTVS+properties">locally convex topological vector spaces</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+convex+topological+vector+space">locally convex topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Banach+space">Banach Spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflexive+Banach+space">reflexive</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Smith+space+%28functional+analysis%29">Smith Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Sobolev+space">Sobolev spaces</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bornological+vector+space">Bornological Vector Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/barrelled+topological+vector+space">Barrelled Vector Spaces</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded</a>, <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded</a>, <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum of an operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-Weierstrass+theorem">Stone-Weierstrass theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></p> </li> </ul> <h2 id="topics_in_functional_analysis">Topics in Functional Analysis</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">Bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theories+in+functional+analysis">Algebraic Theories in Functional Analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">An Elementary Treatment of Hilbert Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+classes+of+Banach+spaces">When are two Banach spaces isomorphic?</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/functional+analysis+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#entries_related_to_analysis'>Entries related to analysis</a></li> <ul> <li><a href='#on_mainstream_analysis'>On mainstream analysis</a></li> <li><a href='#on_foundations'>On foundations</a></li> <li><a href='#on_smoothness_and_generalized_lie_theory'>On smoothness and generalized Lie theory</a></li> <li><a href='#on_geometric_function_theory_and_quantization'>On geometric function theory and quantization</a></li> <li><a href='#on_quantization_and_the_geometry_of_differential_operators'>On quantization and the geometry of differential operators</a></li> <li><a href='#on_contructivism_and_computable_analysis'>On contructivism and computable analysis</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesGeneral'>General</a></li> <li><a href='#ReferencesConstructiveAnalysis'>Constructive analysis</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>, <em>analysis</em> usually refers to any of a broad family of fields that deals with a general theory of <em><a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limits</a></em> in the sense of <a class="existingWikiWord" href="/nlab/show/convergence">convergence</a> of <a class="existingWikiWord" href="/nlab/show/sequences">sequences</a> (or more generally of <a class="existingWikiWord" href="/nlab/show/nets">nets</a>), particularly those fields that pursue developments that originated in “the <a class="existingWikiWord" href="/nlab/show/calculus">calculus</a>”, i.e., the theory of <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> (<a class="existingWikiWord" href="/nlab/show/differential+calculus">differential calculus</a>) and <a class="existingWikiWord" href="/nlab/show/integration">integration</a> (<a class="existingWikiWord" href="/nlab/show/integral+calculus">integral calculus</a>) of <a class="existingWikiWord" href="/nlab/show/real+numbers">real</a> and <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex</a>-valued <a class="existingWikiWord" href="/nlab/show/functions">functions</a>. The classical foundation of this general subject is usually based on the idea that the <a class="existingWikiWord" href="/nlab/show/real+numbers">real number system</a> is describable as a <a class="existingWikiWord" href="/nlab/show/complete+space">complete</a> <a class="existingWikiWord" href="/nlab/show/ordered+field">ordered field</a>, or more generally on the concept of <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a>. Their <a class="existingWikiWord" href="/nlab/show/distance">distance</a> functions allow to formalize concepts like <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuity</a> and <a class="existingWikiWord" href="/nlab/show/convergence">convergence</a> in terms of existence of sufficiently small <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a>. Many concepts of this “<a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic analysis</a>” have equivalent formulations in terms of simple <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> with respect to the <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a> of metric spaces, and this way the field of analysis has a large overlap with the field of <em><a class="existingWikiWord" href="/nlab/show/topology">topology</a></em>, this is particularly true for <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a> and the theory of <a class="existingWikiWord" href="/nlab/show/topological+vector+spaces">topological vector spaces</a>.</p> <p>Analysis can also refer to other responses to the problem of founding these developments, especially “<a class="existingWikiWord" href="/nlab/show/infinitesimal+analysis">infinitesimal analysis</a>” which admits <a class="existingWikiWord" href="/nlab/show/infinitesimal+quantities">infinitesimal quantities</a> not found in the classical real number system and which takes various forms, for example the <a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a> first introduced by <a class="existingWikiWord" href="/nlab/show/Abraham+Robinson">Abraham Robinson</a>, or “<a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential analysis</a>” whose rigorous foundations were largely introduced by <a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a> and other <a class="existingWikiWord" href="/nlab/show/category+theory">category theorists</a> who, following the example of <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, consider <a class="existingWikiWord" href="/nlab/show/nilpotent+infinitesimals">nilpotent infinitesimals</a> (instead of invertible ones à la Robinson) as a basis for understanding <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>.</p> <h2 id="entries_related_to_analysis">Entries related to analysis</h2> <h3 id="on_mainstream_analysis">On mainstream analysis</h3> <p>Some of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>lab entries related to <strong>mathematical analysis</strong> include <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/convergence">convergence</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a>, <a class="existingWikiWord" href="/nlab/show/harmonic+analysis">harmonic analysis</a>, <a class="existingWikiWord" href="/nlab/show/complex+analysis">complex analysis</a>, <a class="existingWikiWord" href="/nlab/show/Weierstrass+preparation+theorem">Weierstrass preparation theorem</a>, <a class="existingWikiWord" href="/nlab/show/several+complex+variables">several complex variables</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/Pontrjagin+dual">Pontrjagin dual</a>, <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/Legendre+polynomial">Legendre polynomial</a>, <a class="existingWikiWord" href="/nlab/show/dilogarithm">dilogarithm</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+algebra">Banach algebra</a>, <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/locally+convex+space">locally convex space</a>, <a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+function">measurable function</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue space</a>, <a class="existingWikiWord" href="/nlab/show/Sobolev+space">Sobolev space</a>, <a class="existingWikiWord" href="/nlab/show/bounded+operator">bounded operator</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact operator</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm operator</a>, <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> (generalized function), <a class="existingWikiWord" href="/nlab/show/hyperfunction">hyperfunction</a>.<a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a>, <a class="existingWikiWord" href="/nlab/show/integral">integral</a>, <a class="existingWikiWord" href="/nlab/show/integration">integration</a>…and a book entry <a class="existingWikiWord" href="/nlab/show/Handbook+of+analysis+and+its+foundations">Handbook of analysis and its foundations</a>. Many of the basic notions used in analysis courses are described in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>lab in the more general <a class="existingWikiWord" href="/nlab/show/topology">topological</a> context if they belong there, e.g. <a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a>, <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a> and so on. Many of the aspects of <a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a> are treated in terms of Riemann surfaces, <a class="existingWikiWord" href="/nlab/show/monodromy">monodromy</a>, <a class="existingWikiWord" href="/nlab/show/local+system">local system</a>s and so on.</p> <h3 id="on_foundations">On foundations</h3> <p>Alternative <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>, especially <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> and those using <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a>, are of traditional interest to the <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> community. For example the <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> of Lawvere and Kock (more in next paragraph) and the <a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a> of Robinson, and its variant, <a class="existingWikiWord" href="/nlab/show/internal+set">internal set</a> theory of Nelson are some of the principal examples. See also <a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a>, <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimal+number">infinitesimal number</a> etc. Many statements are about the versions without the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> and so on; we like to state clean and minimal conditions when possible.</p> <h3 id="on_smoothness_and_generalized_lie_theory">On smoothness and generalized Lie theory</h3> <p>Various smoothness concepts in geometry, rarely studied in standard courses of analysis, but sometimes relevant, were studied to fair extent (and sometimes with innovations) in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>lab. These smoothness concepts are built using some primitive notions in rather generalized (often categorical) setups: <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+differential">Kähler differential</a>, <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>, <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a>, <a class="existingWikiWord" href="/nlab/show/jet+bundles">jet bundles</a>, resolution of diagonal, <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal object</a>, <a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a>, <a class="existingWikiWord" href="/nlab/show/generalized+smooth+algebra">generalized smooth algebra</a>, <a class="existingWikiWord" href="/nlab/show/tangent+category">tangent category</a>, <a class="existingWikiWord" href="/nlab/show/cotangent+complex">cotangent complex</a> as defining ingredients of various notions of smoothness and smooth spaces. Main framework to systematize in geometry similar notion studied in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>lab is <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> but many other examples are also represented. Let us mention <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a>, <a class="existingWikiWord" href="/nlab/show/stratifold">stratifold</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a>, and some graded and super analogues (<a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a>); some concepts of smoothness are rather algebraic, e.g. <a class="existingWikiWord" href="/nlab/show/formal+smoothness">formal smoothness</a> of <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a>; see also <a class="existingWikiWord" href="/nlab/show/algebraic+approaches+to+differential+calculus">algebraic approaches to differential calculus</a>. Special attention in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>lab has been paid to smooth group like objects like <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> and their superanalogues and <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>s, as well as to their tangent structures like <a class="existingWikiWord" href="/nlab/show/Lie+algebroids">Lie algebroids</a> and their interrelations (<a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>: <a class="existingWikiWord" href="/nlab/show/integration">integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>).</p> <h3 id="on_geometric_function_theory_and_quantization">On geometric function theory and quantization</h3> <p>Some other entries are related to the conceptual and categorical understanding of Feynman <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>, however so far from physical, conceptual and formal point of view only (and not of analytic theory). This is closely related to understanding various higher categorical spaces of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> in geometry and in study of sigma-models in physics. This is here called <a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a> (cf. <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a>, <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>…).</p> <h3 id="on_quantization_and_the_geometry_of_differential_operators">On quantization and the geometry of differential operators</h3> <p>Very relevant for <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> is also the geometric study of differential operators (see <a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/diffiety">diffiety</a>) and distributions (cf. <a class="existingWikiWord" href="/nlab/show/microlocal+analysis">microlocal analysis</a>), by analysis of oscillating integrals (<a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a>), <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> (esp. the geometry of <a class="existingWikiWord" href="/nlab/show/lagrangian+submanifold">lagrangian submanifold</a>s which could often be viewed as quantum points) etc. Some of the topological properties of differential operators are studied in <a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, where special role have so called <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a>s. Sometimes it is possible or even useful to avoid fine analysis by using the <a class="existingWikiWord" href="/nlab/show/algebraic+approaches+to+differential+calculus">algebraic approaches to differential calculus</a> and <a class="existingWikiWord" href="/nlab/show/regular+differential+operator">differential operators</a>, what also makes possible some noncommutative analogues.</p> <h3 id="on_contructivism_and_computable_analysis">On contructivism and computable analysis</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a>, <a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+analysis">geometric analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+analysis">infinitesimal analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+analysis">matrix analysis</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="ReferencesGeneral">General</h3> <p>Textbooks accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Apostol">Tom Apostol</a>, <em>Mathematical Analysis</em> (1973) [ISBN:0201002884, <a href="http://webpages.iust.ac.ir/amtehrani/files/Addison%20Wesley%20-%20Mathematical%20Analysis%20_%20Apostol%20%285Th%20Ed%29%20%281981%29.pdf">pdf</a>]</p> </li> <li id="Rudin64"> <p><a class="existingWikiWord" href="/nlab/show/Walter+Rudin">Walter Rudin</a>, <em>Principles of Mathematical Analysis</em>, McGraw-Hill (1964, 1976) [<a href="https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Schechter">Eric Schechter</a>, <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Analysis+and+its+Foundations">Handbook of Analysis and its Foundations</a></em>, Academic Press (1996) (<a href="http://www.math.vanderbilt.edu/~schectex/ccc/">web</a>)</p> </li> </ul> <p>Discussion of the history, amplifying its roots all the way back in <a class="existingWikiWord" href="/nlab/show/Zeno%27s+paradoxes+of+motion">Zeno's paradoxes of motion</a> is in</p> <ul> <li id="Boyer49">Carl Benjamin Boyer, <em>The history of the Calculus and its conceptual development</em>, Dover 1949</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Mathematical_analysis">Mathematical analysis</a></em></li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/calculus">calculus</a></em>.</p> <h3 id="ReferencesConstructiveAnalysis">Constructive analysis</h3> <p>The formulation of <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a> in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, hence <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a>, was maybe initiated in</p> <ul> <li id="Bishop"><a class="existingWikiWord" href="/nlab/show/Errett+Bishop">Errett Bishop</a>, <em>Foundations of constructive analysis.</em> McGraw-Hill, (1967)</li> </ul> <p>together with the basic notion of <a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>/<a class="existingWikiWord" href="/nlab/show/setoid">setoid</a>. Implementations of constructive <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> analysis in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> implemented in <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a> are discussed in</p> <ul> <li> <p>R. O’Connor, <em>A Monadic, Functional Implementation of Real Numbers</em>. MSCS, 17(1):129-159, 2007 (<a href="http://arxiv.org/abs/cs/0605058">arXiv:0605058</a>)</p> </li> <li> <p>R. O’Connor, <em>Certified exact transcendental real number computation in Coq</em>, In TPHOLs 2008, LNCS 5170, pages 246–261, 2008.</p> </li> <li> <p>R. O’Connor, <em>Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory</em>, PhD thesis, Radboud University Nijmegen, 2009.</p> </li> <li> <p>Robbert Krebbers, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>Type classes for efficient exact real arithmetic in Coq</em> (<a href="http://arxiv.org/abs/1106.3448/">arXiv:1106.3448</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a> such as in <a class="existingWikiWord" href="/nlab/show/univalent+foundations">univalent foundations</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>), see the references <a href="https://ncatlab.org/nlab/show/constructive+analysis#References">there</a>, such as:</p> <ul> <li id="Booij20"><a class="existingWikiWord" href="/nlab/show/Auke+Booij">Auke Booij</a>, <em>Analysis in Univalent Type Theory</em> (2020) [<a href="http://etheses.bham.ac.uk/id/eprint/10411">etheses:10411</a>, <a href="https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Booij-AnalysisInUF.pdf" title="pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 5, 2023 at 09:01:43. 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