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<!DOCTYPE html><html><head><title>A Course in Functional Analysis - John B. 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The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts."/><meta property="og:title" content="A Course in Functional Analysis"/><meta property="og:type" content="book"/><meta property="og:site_name" content="Google Books"/><meta property="og:image" content="https://books.google.com.sg/books/content?id=ix4P1e6AkeIC&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE70Gj7YynZ4_WoCx8HR5LwJumt9iyG6v1dSxn4RMbg_7tjM7ORnYq4REwHY5SfYQou0VvjH4UU_VBFntcMN1LjlgFX69S0JMo9PYlcVwaP5l6fLwBzUx-GXL1zVroUM0k4BVCm-f"/><link rel="image_src" href="https://books.google.com.sg/books/content?id=ix4P1e6AkeIC&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE70Gj7YynZ4_WoCx8HR5LwJumt9iyG6v1dSxn4RMbg_7tjM7ORnYq4REwHY5SfYQou0VvjH4UU_VBFntcMN1LjlgFX69S0JMo9PYlcVwaP5l6fLwBzUx-GXL1zVroUM0k4BVCm-f"/><script></script><style>#gbar,#guser{font-size:13px;padding-top:1px !important;}#gbar{height:22px}#guser{padding-bottom:7px !important;text-align:right}.gbh,.gbd{border-top:1px solid #c9d7f1;font-size:1px}.gbh{height:0;position:absolute;top:24px;width:100%}@media all{.gb1{height:22px;margin-right:.5em;vertical-align:top}#gbar{float:left}}a.gb1,a.gb4{text-decoration:underline !important}a.gb1,a.gb4{color:#00c !important}.gbi .gb4{color:#dd8e27 !important}.gbf .gb4{color:#900 !important} #gbar { padding:.3em .6em !important;}</style></head><body class=""><div id=gbar><nobr><a target=_blank class=gb1 href="https://www.google.com.sg/search?tab=pw">Search</a> <a target=_blank class=gb1 href="https://www.google.com.sg/imghp?hl=en&tab=pi">Images</a> <a target=_blank class=gb1 href="https://maps.google.com.sg/maps?hl=en&tab=pl">Maps</a> <a target=_blank class=gb1 href="https://play.google.com/?hl=en&tab=p8">Play</a> <a target=_blank class=gb1 href="https://www.youtube.com/?tab=p1">YouTube</a> <a target=_blank class=gb1 href="https://news.google.com/?tab=pn">News</a> <a target=_blank class=gb1 href="https://mail.google.com/mail/?tab=pm">Gmail</a> <a target=_blank class=gb1 href="https://drive.google.com/?tab=po">Drive</a> <a target=_blank class=gb1 style="text-decoration:none" href="https://www.google.com.sg/intl/en/about/products?tab=ph"><u>More</u> »</a></nobr></div><div id=guser width=100%><nobr><span id=gbn class=gbi></span><span id=gbf class=gbf></span><span id=gbe></span><a target=_top id=gb_70 href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fid%3Dix4P1e6AkeIC%26redir_esc%3Dy%26hl%3Den&hl=en&ec=GAZACg" class=gb4>Sign in</a></nobr></div><div class=gbh style=left:0></div><div class=gbh style=right:0></div><div role="alert" style="position: absolute; left: 0; right: 0;"><a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&redir_esc=y&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. 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Conway</span></a></div><div><span dir=ltr>Springer Science & Business Media</span>, <span dir=ltr>25 Jan 1994</span> - <a class="secondary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22&source=gbs_ge_summary_r&cad=0"><span dir=ltr>Mathematics</span></a> - <span dir=ltr>400 pages</span></div></div><div id=synopsis><div id=synopsis-window><div id=synopsistext dir=ltr class="sa">Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts.</div></div></div><div class="search_box_wrapper"><form action=/books id=search_form style="margin:0px;padding:0px;" method=get> <input type=hidden name="redir_esc" value="y"><input type=hidden name="id" value="ix4P1e6AkeIC"><table cellpadding=0 cellspacing=0 class="swv-table"><tr><td class="swv-td-search"><span><input id=search_form_input type=text maxlength=1024 class="text_flat swv-input-search" aria-label="Search in this book" name=q value="" title="Search inside" accesskey=i></span></td><td class="swv-td-space"><div> </div></td><td><input type=submit value="Search inside"></td></tr></table><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script></form><div id="preview-link"><a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&printsec=frontcover" class="primary"><span dir=ltr>Preview this book</span> »</a></div></div></td> </tr></table><div id="summary-second-column"></div></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="toc_anchor"></a>Contents</h3><div id=toc class=about_content><div id=toc_v><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Hilbert Spaces</span></span></div></td><td class="toc_number" align=right>1</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Orthogonality</span></span></div></td><td class="toc_number" align=right>7</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 The Riesz Representation Theorem</span></span></div></td><td class="toc_number" align=right>11</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Orthonormal Sets of Vectors and Bases</span></span></div></td><td class="toc_number" align=right>14</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Isomorphic Hilbert Spaces and the Fourier Transform for the Circle</span></span></div></td><td class="toc_number" align=right>19</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>6 The Direct Sum of Hilbert Spaces</span></span></div></td><td class="toc_number" align=right>23</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Operators on Hilbert Space</span></span></div></td><td class="toc_number" align=right>26</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 The Adjoint of an Operator</span></span></div></td><td class="toc_number" align=right>31</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 Compact Operators</span></span></div></td><td class="toc_number" align=right>173</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Invariant Subspaces</span></span></div></td><td class="toc_number" align=right>178</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Weakly Compact Operators</span></span></div></td><td class="toc_number" align=right>183</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Banach Algebras and Spectral Theory for Operators on a Banach Space</span></span></div></td><td class="toc_number" align=right>187</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Ideals and Quotients</span></span></div></td><td class="toc_number" align=right>191</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 The Spectrum</span></span></div></td><td class="toc_number" align=right>195</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 The Riesz Functional Calculus</span></span></div></td><td class="toc_number" align=right>199</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Dependence of the Spectrum on the Algebra</span></span></div></td><td class="toc_number" align=right>205</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/></div><span onclick="_OC_setListSectionVisible('toc_h', 1)" class=morelesslink id=toc_hc0 style="display:none"><br>More</span><div id=toc_hd1><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 Projections and Idempotents Invariant and Reducing Subspaces</span></span></div></td><td class="toc_number" align=right>36</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Compact Operators</span></span></div></td><td class="toc_number" align=right>41</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 The Diagonalization of Compact SelfAdjoint Operators</span></span></div></td><td class="toc_number" align=right>46</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>SturmLiouville Systems</span></span></div></td><td class="toc_number" align=right>49</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>7 The Spectral Theorem and Functional Calculus for Compact Normal Operators</span></span></div></td><td class="toc_number" align=right>54</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>8 Unitary Equivalence for Compact Normal Operators</span></span></div></td><td class="toc_number" align=right>60</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Banach Spaces</span></span></div></td><td class="toc_number" align=right>63</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Linear Operators on Normed Spaces</span></span></div></td><td class="toc_number" align=right>67</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 Finite Dimensional Normed Spaces</span></span></div></td><td class="toc_number" align=right>69</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Quotients and Products of Normed Spaces</span></span></div></td><td class="toc_number" align=right>70</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Linear Functionals</span></span></div></td><td class="toc_number" align=right>73</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>6 The HahnBanach Theorem</span></span></div></td><td class="toc_number" align=right>77</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Banach Limits</span></span></div></td><td class="toc_number" align=right>82</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Runges Theorem</span></span></div></td><td class="toc_number" align=right>83</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Ordered Vector Spaces</span></span></div></td><td class="toc_number" align=right>86</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>10 The Dual of a Quotient Space and a Subspace</span></span></div></td><td class="toc_number" align=right>88</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>11 Reflexive Spaces</span></span></div></td><td class="toc_number" align=right>89</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>12 The Open Mapping and Closed Graph Theorems</span></span></div></td><td class="toc_number" align=right>90</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>13 Complemented Subspaces of a Banach Space</span></span></div></td><td class="toc_number" align=right>93</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>14 The Principle of Uniform Boundedness</span></span></div></td><td class="toc_number" align=right>95</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Locally Convex Spaces</span></span></div></td><td class="toc_number" align=right>99</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Metrizable and Normable Locally Convex Spaces</span></span></div></td><td class="toc_number" align=right>105</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 Some Geometric Consequences of the HahnBanach Theorem</span></span></div></td><td class="toc_number" align=right>108</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Some Examples of the Dual Space of a Locally Convex Space</span></span></div></td><td class="toc_number" align=right>114</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Inductive Limits and the Space of Distributions</span></span></div></td><td class="toc_number" align=right>116</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Weak Topologies</span></span></div></td><td class="toc_number" align=right>124</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 The Dual of a Subspace and a Quotient Space</span></span></div></td><td class="toc_number" align=right>128</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 Alaoglus Theorem</span></span></div></td><td class="toc_number" align=right>130</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Reflexivity Revisited</span></span></div></td><td class="toc_number" align=right>131</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Separability and Metrizability</span></span></div></td><td class="toc_number" align=right>134</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The StoneCech Compactification</span></span></div></td><td class="toc_number" align=right>137</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>7 The KreinMilman Theorem</span></span></div></td><td class="toc_number" align=right>141</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The StoneWeierstrass Theorem</span></span></div></td><td class="toc_number" align=right>145</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>9 The Schauder Fixed Point Theorem</span></span></div></td><td class="toc_number" align=right>149</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>10 The RyllNardzewski Fixed Point Theorem</span></span></div></td><td class="toc_number" align=right>151</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Haar Measure on a Compact Group</span></span></div></td><td class="toc_number" align=right>154</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>12 The KreinSmulian Theorem</span></span></div></td><td class="toc_number" align=right>159</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>13 Weak Compactness</span></span></div></td><td class="toc_number" align=right>163</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Linear Operators on a Banach Space</span></span></div></td><td class="toc_number" align=right>166</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 The BanachStone Theorem</span></span></div></td><td class="toc_number" align=right>171</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>6 The Spectrum of a Linear Operator</span></span></div></td><td class="toc_number" align=right>208</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>7 The Spectral Theory of a Compact Operator</span></span></div></td><td class="toc_number" align=right>214</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>8 Abelian Banach Algebras</span></span></div></td><td class="toc_number" align=right>218</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>9 The Group Algebra of a Locally Compact Abelian Group</span></span></div></td><td class="toc_number" align=right>223</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>CAlgebras</span></span></div></td><td class="toc_number" align=right>232</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Abelian CAlgebras and the Functional Calculus in CAlgebras</span></span></div></td><td class="toc_number" align=right>236</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 The Positive Elements in a CAlgebra</span></span></div></td><td class="toc_number" align=right>240</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Ideals and Quotients of CAlgebras</span></span></div></td><td class="toc_number" align=right>245</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Representations of CAlgebras and the GelfandNaimarkSegal Construction</span></span></div></td><td class="toc_number" align=right>248</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Normal Operators on Hilbert Space</span></span></div></td><td class="toc_number" align=right>255</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 The Spectral Theorem</span></span></div></td><td class="toc_number" align=right>262</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 StarCyclic Normal Operators</span></span></div></td><td class="toc_number" align=right>268</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Some Applications of the Spectral Theorem</span></span></div></td><td class="toc_number" align=right>271</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Topologies on B H</span></span></div></td><td class="toc_number" align=right>274</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>6 Commuting Operators</span></span></div></td><td class="toc_number" align=right>276</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>7 Abelian von Neumann Algebras</span></span></div></td><td class="toc_number" align=right>281</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Conclusion of the Saga</span></span></div></td><td class="toc_number" align=right>285</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>9 Invariant Subspaces for Normal Operators</span></span></div></td><td class="toc_number" align=right>290</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A Complete Set of Unitary Invariants</span></span></div></td><td class="toc_number" align=right>293</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Unbounded Operators</span></span></div></td><td class="toc_number" align=right>303</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Symmetric and SelfAdjoint Operators</span></span></div></td><td class="toc_number" align=right>308</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 The Cayley Transform</span></span></div></td><td class="toc_number" align=right>316</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 Unbounded Normal Operators and the Spectral Theorem</span></span></div></td><td class="toc_number" align=right>319</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 Stones Theorem</span></span></div></td><td class="toc_number" align=right>327</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>6 The Fourier Transform and Differentiation</span></span></div></td><td class="toc_number" align=right>334</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>7 Moments</span></span></div></td><td class="toc_number" align=right>343</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Fredholm Theory</span></span></div></td><td class="toc_number" align=right>347</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Fredholm Operators</span></span></div></td><td class="toc_number" align=right>349</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>3 Fredholm Theory</span></span></div></td><td class="toc_number" align=right>352</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>4 The Essential Spectrum</span></span></div></td><td class="toc_number" align=right>358</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>5 The Components of J F</span></span></div></td><td class="toc_number" align=right>362</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>6 A Finer Analysis of the Spectrum</span></span></div></td><td class="toc_number" align=right>363</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Preliminaries</span></span></div></td><td class="toc_number" align=right>369</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>2 Topology</span></span></div></td><td class="toc_number" align=right>371</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Dual of Lp𝜇</span></span></div></td><td class="toc_number" align=right>375</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Dual of C₀X</span></span></div></td><td class="toc_number" align=right>378</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Bibliography</span></span></div></td><td class="toc_number" align=right>384</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>List of Symbols</span></span></div></td><td class="toc_number" align=right>391</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&pg=PA395&source=gbs_toc_r&cad=1" ><span title="Index" style="white-space:nowrap"><span dir=ltr>Index</span></span></a></div></td><td class="toc_number" align=right>395</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&printsec=copyright" ><span title="Copyright" style="white-space:nowrap"><span dir=ltr>Copyright</span></span></a></div></td><td class="toc_number" align=right></td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/><span onclick="_OC_setListSectionVisible('toc_h', 0)" class=morelesslink id=toc_hc1 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Conway</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2013</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=wWS-DwAAQBAJ&source=gbs_book_other_versions_r&cad=2" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=wWS-DwAAQBAJ&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=wWS-DwAAQBAJ&printsec=frontcover&source=gbs_book_other_versions_r&cad=2"><span dir=ltr>A Course in Functional Analysis</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22John+B+Conway%22" class="secondary"><span dir=ltr>John B Conway</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2019</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=Z9JFAQAAIAAJ&source=gbs_book_other_versions_r&cad=2" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=Z9JFAQAAIAAJ&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=Z9JFAQAAIAAJ&source=gbs_book_other_versions_r&cad=2"><span dir=ltr>A Course in Functional Analysis</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22John+B.+Conway%22" class="secondary"><span dir=ltr>John B. Conway</span></a></span><br/><span><span style="color:#999">Snippet view</span> - 1985</span><br/></span></div></td><td align=right></td></tr></table></div></div><div style="clear:both"></div><div class="showall"><a class="secondary" href="https://books.google.com.sg/books?q=editions:ISBN0387972455&id=ix4P1e6AkeIC&source=gbs_book_other_versions_r&cad=2">View all »</a></div><script>(function () {var fn = window['_OC_WSBookList'] || window['_OC_BookList'];fn && fn('book_other_versions', [{"title":"A Course in Functional Analysis","authors":"John B. 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10.5px;}.cloud7 {color: #6057CC;font-size: 11px;}.cloud6 {color: #574BCC;font-size: 11.5px;}.cloud5 {color: #4E3DCC;font-size: 12px;}.cloud4 {color: #4632CC;font-size: 14px;}.cloud3 {color: #3D26CC;font-size: 16px;}.cloud2 {color: #341ACC;font-size: 18px;}.cloud1 {color: #2B0DCC;font-size: 20px;}.cloud0 {color: #2200CC;font-size: 22px;}.cloud {margin-top: 4px;line-height: 24px;}.cloud a {margin-right: 6px;text-decoration: none;}.cloud a:hover {text-decoration: underline;}</style><div class=cloud><a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=A%E2%82%81&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>A₁</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=abelian&source=gbs_word_cloud_r&cad=3" class="cloud6"><span dir=ltr>abelian</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=AEB(H&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>AEB(H</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=assume&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>assume</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=ball&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>ball</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Banach+algebra&source=gbs_word_cloud_r&cad=3" class="cloud2"><span dir=ltr>Banach algebra</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Banach+space&source=gbs_word_cloud_r&cad=3" class="cloud1"><span dir=ltr>Banach space</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Borel&source=gbs_word_cloud_r&cad=3" class="cloud6"><span dir=ltr>Borel</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=bounded+linear&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>bounded linear</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=bounded+operator&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>bounded operator</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=C*-algebra&source=gbs_word_cloud_r&cad=3" class="cloud1"><span dir=ltr>C*-algebra</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=closed&source=gbs_word_cloud_r&cad=3" class="cloud6"><span dir=ltr>closed</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Co(X&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>Co(X</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=compact+normal+operator&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>compact normal operator</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=compact+operator&source=gbs_word_cloud_r&cad=3" class="cloud3"><span dir=ltr>compact operator</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=compact+subset&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>compact subset</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=continuous+function&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>continuous function</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=converges&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>converges</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=convex+subset&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>convex subset</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Corollary&source=gbs_word_cloud_r&cad=3" class="cloud5"><span dir=ltr>Corollary</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=countable&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>countable</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=defined&source=gbs_word_cloud_r&cad=3" class="cloud3"><span dir=ltr>defined</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Definition&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>Definition</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=denoted&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>denoted</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=dense&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>dense</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=dim+ker&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>dim ker</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Example&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>Example</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Exercise&source=gbs_word_cloud_r&cad=3" class="cloud5"><span dir=ltr>Exercise</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=extreme+point&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>extreme point</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=fact&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>fact</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=finite+dimensional&source=gbs_word_cloud_r&cad=3" class="cloud6"><span dir=ltr>finite dimensional</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Fredholm&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>Fredholm</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=functional+calculus&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>functional calculus</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=functions+f&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>functions f</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=h+in+H&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>h in H</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=h%E2%82%81&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>h₁</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=h%E2%82%82&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>h₂</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Hence&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>Hence</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Hilbert+space&source=gbs_word_cloud_r&cad=3" class="cloud1"><span dir=ltr>Hilbert space</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=homomorphism&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>homomorphism</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=implies&source=gbs_word_cloud_r&cad=3" class="cloud6"><span dir=ltr>implies</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=invariant+subspace&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>invariant subspace</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=invertible&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>invertible</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=isometrically+isomorphic&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>isometrically isomorphic</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=isometry&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>isometry</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=ker(A&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>ker(A</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=L%C2%B2(%C2%B5&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>L²(µ</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Lemma&source=gbs_word_cloud_r&cad=3" class="cloud5"><span dir=ltr>Lemma</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=linear+functional&source=gbs_word_cloud_r&cad=3" class="cloud6"><span dir=ltr>linear functional</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=linear+manifold&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>linear manifold</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=locally+compact&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>locally compact</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Math&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>Math</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=measure+space&source=gbs_word_cloud_r&cad=3" class="cloud3"><span dir=ltr>measure space</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=metric&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>metric</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=normed+space&source=gbs_word_cloud_r&cad=3" class="cloud3"><span dir=ltr>normed space</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Note&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>Note</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=orthogonal&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>orthogonal</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=orthonormal&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>orthonormal</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=polynomial&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>polynomial</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=PROOF&source=gbs_word_cloud_r&cad=3" class="cloud3"><span dir=ltr>PROOF</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Prove&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>Prove</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=ran(A&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>ran(A</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=reader&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>reader</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=reflexive&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>reflexive</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=representation&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>representation</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=result&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>result</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=self-adjoint+operator&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>self-adjoint operator</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=semi-Fredholm&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>semi-Fredholm</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=semi-Fredholm+operator&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>semi-Fredholm operator</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=seminorm&source=gbs_word_cloud_r&cad=3" class="cloud3"><span dir=ltr>seminorm</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=separable&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>separable</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=sequence&source=gbs_word_cloud_r&cad=3" class="cloud6"><span dir=ltr>sequence</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=space+and+let&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>space and let</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=spectral+measure&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>spectral measure</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Spectral+Theorem&source=gbs_word_cloud_r&cad=3" class="cloud2"><span dir=ltr>Spectral Theorem</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=statements+are+equivalent&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>statements are equivalent</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Suppose&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>Suppose</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=surjective&source=gbs_word_cloud_r&cad=3" class="cloud4"><span dir=ltr>surjective</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=symmetric+operator&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>symmetric operator</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Theory&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>Theory</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=topology&source=gbs_word_cloud_r&cad=3" class="cloud3"><span dir=ltr>topology</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=unique&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>unique</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=unitary&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>unitary</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=vector+space&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>vector space</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=Verify+the+statements&source=gbs_word_cloud_r&cad=3" class="cloud9"><span dir=ltr>Verify the statements</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=weak-star&source=gbs_word_cloud_r&cad=3" class="cloud7"><span dir=ltr>weak-star</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=weakly+compact&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>weakly compact</span></a> <a href="https://books.google.com.sg/books?id=ix4P1e6AkeIC&q=y%E2%82%81&source=gbs_word_cloud_r&cad=3" class="cloud8"><span dir=ltr>y₁</span></a></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="citations_module_anchor"></a>References to this book</h3><div id=citations_module class=about_content><div id=citations_module_v><div class="left-column-outer"><div class="left-column-inner"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=MbFBXyuxLKgC&source=gbs_citations_module_r&cad=4" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=MbFBXyuxLKgC&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=MbFBXyuxLKgC&printsec=references&vq=%22A+Course+in+Functional+Analysis%22&source=gbs_citations_module_r&cad=4"><span dir=ltr>Loop Groups</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Andrew+Pressley%22" class="secondary"><span dir=ltr>Andrew Pressley</span></a>,<a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Graeme+Segal%22" class="secondary"><span dir=ltr>Graeme Segal</span></a></span><br/><span><span style="color:#999">No preview available</span> - 1988</span><br/></span></div></td><td align=right></td></tr></table></div><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=Jqk-TYTmtMYC&source=gbs_citations_module_r&cad=4" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=Jqk-TYTmtMYC&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=Jqk-TYTmtMYC&printsec=frontcover&vq=%22A+Course+in+Functional+Analysis%22&source=gbs_citations_module_r&cad=4"><span dir=ltr>Mathematical Control Theory: Deterministic Finite Dimensional Systems</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Eduardo+D.+Sontag%22" class="secondary"><span dir=ltr>Eduardo D. Sontag</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 1998</span><br/></span></div></td><td align=right></td></tr></table></div><a class="secondary" href="https://books.google.com.sg/books?q=cites:ISBN0387972455&id=ix4P1e6AkeIC&source=gbs_citations_module_r&cad=4"><span dir=ltr>All Book Search results &raquo;</span></a></div></div><div style="clear:both"></div> </div></div></div><div class=vertical_module_list_row><h3 class="about_title">Bibliographic information</h3><div class="about_content" id="metadata_content" style="padding-bottom:.3em"><div class="metadata_sectionwrap"><table id="metadata_content_table"><tr class="metadata_row"><td class="metadata_label">Title</td><td class="metadata_value"><span dir=ltr>A Course in Functional Analysis</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Graduate+Texts+in+Mathematics%22&source=gbs_metadata_r&cad=5"><i><span dir=ltr>Volume 96 of Graduate Texts in Mathematics</span></i></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Author</span></td><td class="metadata_value"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22John+B.+Conway%22&source=gbs_metadata_r&cad=5"><span dir=ltr>John B. Conway</span></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Edition</span></td><td class="metadata_value"><span dir=ltr>2, illustrated, reprint</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>Springer Science & Business Media, 1994</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0387972455, 9780387972459</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>400 pages</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Subjects</span></td><td class="metadata_value"><div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22" itemprop="url" dir=ltr><span itemprop="title">Mathematics</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Mathematical+Analysis%22" itemprop="url" dir=ltr><span itemprop="title">Mathematical Analysis</span></a></div><br><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Calculus%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Calculus</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Mathematical+Analysis%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Mathematical Analysis</span></a></td></tr><tr class="metadata_row"><td>&nbsp</td><td>&nbsp</td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Export Citation</span></td><td class="metadata_value"><a class="gb-button " href="https://books.google.com.sg/books/download/A_Course_in_Functional_Analysis.bibtex?id=ix4P1e6AkeIC&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/A_Course_in_Functional_Analysis.enw?id=ix4P1e6AkeIC&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/A_Course_in_Functional_Analysis.ris?id=ix4P1e6AkeIC&output=ris"><span dir=ltr>RefMan</span></a></td></tr></table></div><div style="clear:both"></div></div></div><script>_OC_addFlags({Host:"https://books.google.com.sg/", IsBooksUnifiedLeftNavEnabled:1, IsZipitFolderCollectionEnabled:1, IsBrowsingHistoryEnabled:1, IsBooksRentalEnabled:1});_OC_Run({"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3Dix4P1e6AkeIC%26redir_esc%3Dy%26hl%3Den\u0026hl=en"}, {"volume_id":"ix4P1e6AkeIC","is_ebook":false,"volumeresult":{"has_flowing_text":false,"has_scanned_text":true,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false},"sample_url":"https://play.google.com/books/reader?id=ix4P1e6AkeIC\u0026source=gbs_atb_hover","is_browsable":true,"is_public_domain":false}, {});</script><div id="footer_table" style="font-size:83%;text-align:center;position:relative;top:20px;height:4.5em;margin-top:2em"><div style="margin-bottom:8px"><a href="/intl/en/googlebooks/about.html"><nobr><nobr>About Google Books</nobr></nobr></a> - <a href="/intl/en/googlebooks/privacy.html"><nobr><nobr>Privacy Policy</nobr></nobr></a> - <a href="/intl/en/googlebooks/tos.html"><nobr><nobr>Terms of Service</nobr></nobr></a> - <a href="http://books.google.com.sg/support/partner/?hl=en-SG"><nobr><nobr>Information for Publishers</nobr></nobr></a> - <a href="http://books.google.com.sg/support/answer/180577?hl=en-SG&url=https://books.google.com.sg/books?id=ix4P1e6AkeIC&redir_esc=y&hl=en&v=ix4P1e6AkeIC&is=atb"><nobr><nobr>Report an issue</nobr></nobr></a> - <a href="http://books.google.com.sg/support/topic/4359341?hl=en-SG"><nobr><nobr>Help</nobr></nobr></a> - <a href="https://www.google.com.sg/"><nobr><nobr>Google Home</nobr></nobr></a></div></div></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>