https://citizendium.org/wiki/index.php?title=Banach_space&feed=atom&action=history
Banach space - Revision history
2024-03-29T07:08:23Z
Revision history for this page on the wiki
MediaWiki 1.39.5
https://citizendium.org/wiki/index.php?title=Banach_space&diff=360321&oldid=prev
imported>Jitse Niesen: move Yoshida ref to Bibliography subpage
2008-07-14T18:13:28Z
<p>move Yoshida ref to Bibliography subpage</p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:13, 14 July 2008</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l23">Line 23:</td>
<td colspan="2" class="diff-lineno">Line 23:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle p\,=\,\infty</math>. The case ''p'' = 2 is special since it is also a [[Hilbert space]] and is in fact the only Hilbert space among the <math>\scriptstyle L^p(\mathbb{T})</math> spaces, <math> \scriptstyle 1\,\leq p\,\leq \infty</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle p\,=\,\infty</math>. The case ''p'' = 2 is special since it is also a [[Hilbert space]] and is in fact the only Hilbert space among the <math>\scriptstyle L^p(\mathbb{T})</math> spaces, <math> \scriptstyle 1\,\leq p\,\leq \infty</math>.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Further reading==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
imported>Jitse Niesen
https://citizendium.org/wiki/index.php?title=Banach_space&diff=360323&oldid=prev
imported>Hendra I. Nurdin: /* Examples of Banach spaces */
2007-12-18T11:05:39Z
<p><span dir="auto"><span class="autocomment">Examples of Banach spaces</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:05, 18 December 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1. The Euclidean space <math>\scriptstyle \mathbb{R}^n</math> with any [[norm (mathematics)|norm]] is a Banach space. More generally, any finite dimensional normed space is a Banach space (due to its isomorphism to some Euclidean space).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1. The Euclidean space <math>\scriptstyle \mathbb{R}^n</math> with any [[norm (mathematics)|norm]] is a Banach space. More generally, any finite dimensional normed space is a Banach space (due to its isomorphism to some Euclidean space).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>2. Let <math>\scriptstyle L^p(\mathbb{T})</math>, <math>\scriptstyle 1\, \leq p \,\leq\, \infty</math>, denote the space of all [[complex number|complex]]-valued measurable <del style="font-weight: bold; text-decoration: none;">function </del>on the unit circle <math>\scriptstyle \mathbb{T}\,=\,\{z \in \mathbb{C} \mid |z|\,=\,1\}</math> of the complex plane (with respect to the [[Haar measure]] <math>\scriptstyle \mu</math> on <math>\scriptstyle \mathbb{T}</math>) satisfying: </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>2. Let <math>\scriptstyle L^p(\mathbb{T})</math>, <math>\scriptstyle 1\, \leq p \,\leq\, \infty</math>, denote the space of all [[complex number|complex]]-valued measurable <ins style="font-weight: bold; text-decoration: none;">functions </ins>on the unit circle <math>\scriptstyle \mathbb{T}\,=\,\{z \in \mathbb{C} \mid |z|\,=\,1\}</math> of the complex plane (with respect to the [[Haar measure]] <math>\scriptstyle \mu</math> on <math>\scriptstyle \mathbb{T}</math>) satisfying: </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math> \int_{\mathbb{T}}|f(z)|^p\,\mu(dz)<\infty</math>,</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math> \int_{\mathbb{T}}|f(z)|^p\,\mu(dz)<\infty</math>,</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
</table>
imported>Hendra I. Nurdin
https://citizendium.org/wiki/index.php?title=Banach_space&diff=360324&oldid=prev
imported>Hendra I. Nurdin at 09:41, 18 December 2007
2007-12-18T09:41:52Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 04:41, 18 December 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16">Line 16:</td>
<td colspan="2" class="diff-lineno">Line 16:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle p\,=\,\infty</math>. Then <math>\scriptstyle L^p(\mathbb{T})</math> is a Banach space with a norm <math>\scriptstyle \|\cdot \|_p</math> defined by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle p\,=\,\infty</math>. Then <math>\scriptstyle L^p(\mathbb{T})</math> is a Banach space with a norm <math>\scriptstyle \|\cdot \|_p</math> defined by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<math> \|f\|_p=\left(\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)\right^{1/p}</math>,</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<math> \|f\|_p=\left(\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)\right<ins style="font-weight: bold; text-decoration: none;">)</ins>^{1/p}</math>,</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle 1\,\leq\, p < \infty </math>, or </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle 1\,\leq\, p < \infty </math>, or </div></td></tr>
</table>
imported>Hendra I. Nurdin
https://citizendium.org/wiki/index.php?title=Banach_space&diff=360325&oldid=prev
imported>Hendra I. Nurdin: /* Examples of Banach spaces */
2007-12-18T09:41:05Z
<p><span dir="auto"><span class="autocomment">Examples of Banach spaces</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 04:41, 18 December 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16">Line 16:</td>
<td colspan="2" class="diff-lineno">Line 16:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle p\,=\,\infty</math>. Then <math>\scriptstyle L^p(\mathbb{T})</math> is a Banach space with a norm <math>\scriptstyle \|\cdot \|_p</math> defined by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle p\,=\,\infty</math>. Then <math>\scriptstyle L^p(\mathbb{T})</math> is a Banach space with a norm <math>\scriptstyle \|\cdot \|_p</math> defined by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<math> \|f\|_p=\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)</math>,</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<math> \|f\|_p=<ins style="font-weight: bold; text-decoration: none;">\left(</ins>\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)<ins style="font-weight: bold; text-decoration: none;">\right^{1/p}</ins></math>,</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle 1\,\leq\, p < \infty </math>, or </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>if <math>\scriptstyle 1\,\leq\, p < \infty </math>, or </div></td></tr>
</table>
imported>Hendra I. Nurdin
https://citizendium.org/wiki/index.php?title=Banach_space&diff=360326&oldid=prev
imported>Hendra I. Nurdin at 08:44, 18 December 2007
2007-12-18T08:44:41Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:44, 18 December 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2">Line 2:</td>
<td colspan="2" class="diff-lineno">Line 2:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In [[mathematics]], particularly in the branch known as [[functional analysis]], a '''Banach space''' is a [[completeness|complete]] [[normed space]]. It is named after famed Hungarian-Polish mathematician [[Stefan Banach]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In [[mathematics]], particularly in the branch known as [[functional analysis]], a '''Banach space''' is a [[completeness|complete]] [[normed space]]. It is named after famed Hungarian-Polish mathematician [[Stefan Banach]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The space of all [[continuous function|continous]] complex (resp. real) linear functionals of a complex (resp. real) Banach space is called its [[dual space]]. This dual space is also a Banach space <del style="font-weight: bold; text-decoration: none;">under </del>when endowed with the operator norm on the continuous (hence, bounded) linear functionals. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The space of all [[continuous function|continous]] complex (resp. real) linear functionals of a complex (resp. real) Banach space is called its [[dual space]]. This dual space is also a Banach space when endowed with the operator norm on the continuous (hence, bounded) linear functionals. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples of Banach spaces==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples of Banach spaces==</div></td></tr>
</table>
imported>Hendra I. Nurdin
https://citizendium.org/wiki/index.php?title=Banach_space&diff=360322&oldid=prev
imported>Hendra I. Nurdin: New
2007-12-18T08:25:20Z
<p>New</p>
<p><b>New page</b></p><div>{{subpages}}<br />
In [[mathematics]], particularly in the branch known as [[functional analysis]], a '''Banach space''' is a [[completeness|complete]] [[normed space]]. It is named after famed Hungarian-Polish mathematician [[Stefan Banach]].<br />
<br />
The space of all [[continuous function|continous]] complex (resp. real) linear functionals of a complex (resp. real) Banach space is called its [[dual space]]. This dual space is also a Banach space under when endowed with the operator norm on the continuous (hence, bounded) linear functionals. <br />
<br />
==Examples of Banach spaces==<br />
1. The Euclidean space <math>\scriptstyle \mathbb{R}^n</math> with any [[norm (mathematics)|norm]] is a Banach space. More generally, any finite dimensional normed space is a Banach space (due to its isomorphism to some Euclidean space).<br />
<br />
2. Let <math>\scriptstyle L^p(\mathbb{T})</math>, <math>\scriptstyle 1\, \leq p \,\leq\, \infty</math>, denote the space of all [[complex number|complex]]-valued measurable function on the unit circle <math>\scriptstyle \mathbb{T}\,=\,\{z \in \mathbb{C} \mid |z|\,=\,1\}</math> of the complex plane (with respect to the [[Haar measure]] <math>\scriptstyle \mu</math> on <math>\scriptstyle \mathbb{T}</math>) satisfying: <br />
:<math> \int_{\mathbb{T}}|f(z)|^p\,\mu(dz)<\infty</math>,<br />
<br />
if <math>\scriptstyle 1\,\leq p\, < \infty </math>, or<br />
<br />
:<math>\mathop{{\rm ess} \sup}_{z \in \mathbb{T}}|f(z)|<\infty,</math><br />
<br />
if <math>\scriptstyle p\,=\,\infty</math>. Then <math>\scriptstyle L^p(\mathbb{T})</math> is a Banach space with a norm <math>\scriptstyle \|\cdot \|_p</math> defined by<br />
<br />
:<math> \|f\|_p=\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)</math>,<br />
<br />
if <math>\scriptstyle 1\,\leq\, p < \infty </math>, or <br />
<br />
:<math>\|f\|_{\infty}=\mathop{{\rm ess} \sup}_{z \in \mathbb{T}}|f(z)|,</math><br />
<br />
if <math>\scriptstyle p\,=\,\infty</math>. The case ''p'' = 2 is special since it is also a [[Hilbert space]] and is in fact the only Hilbert space among the <math>\scriptstyle L^p(\mathbb{T})</math> spaces, <math> \scriptstyle 1\,\leq p\,\leq \infty</math>.<br />
<br />
==Further reading==<br />
1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980</div>
imported>Hendra I. Nurdin