Generalized Tenney dual norms and Tp tuning space: Difference between revisions
Wikispaces>genewardsmith **Imported revision 515964958 - Original comment: ** |
Wikispaces>clumma **Imported revision 535152546 - Original comment: ** |
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:clumma|clumma]] and made on <tt>2014-12-14 22:34:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>535152546</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Dual Norms= | ||
=Dual Norms= | |||
Given any [[Generalized Tenney Norms and Tp Interval Space|Tp norm]] on an interval space **Tp<span style="font-size: 10px; vertical-align: super;">G</span>** associated with a group **G**, we can define a corresponding **dual Tq* norm** on the dual space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** which satisfies the following identity: | Given any [[Generalized Tenney Norms and Tp Interval Space|Tp norm]] on an interval space **Tp<span style="font-size: 10px; vertical-align: super;">G</span>** associated with a group **G**, we can define a corresponding **dual Tq* norm** on the dual space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** which satisfies the following identity: | ||
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Note that this is the quotient norm induced on the space **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**), where ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the [[@http://www.math.unl.edu/~s-bbockel1/928/node25.html|Hahn-Banach theorem]], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space **Tp<span style="font-size: 10px; vertical-align: super;">L</span>** and our subspace is **Tp<span style="font-size: 10px; vertical-align: super;">G</span>**, this proves that our dual space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** must be isometrically isomorphic to **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**).</pre></div> | Note that this is the quotient norm induced on the space **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**), where ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the [[@http://www.math.unl.edu/~s-bbockel1/928/node25.html|Hahn-Banach theorem]], which demonstrates that the dual space M* to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V*/ker(M), where ker(M) is the set of all f in V* such that f(M) = 0. Since our vector space **Tp<span style="font-size: 10px; vertical-align: super;">L</span>** and our subspace is **Tp<span style="font-size: 10px; vertical-align: super;">G</span>**, this proves that our dual space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** must be isometrically isomorphic to **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**).</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Generalized Tenney Dual Norms and Tp Tuning Space</title></head><body> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Generalized Tenney Dual Norms and Tp Tuning Space</title></head><body><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Dual Norms"></a><!-- ws:end:WikiTextHeadingRule:5 -->Dual Norms</h1> | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Dual Norms"></a><!-- ws:end:WikiTextHeadingRule:5 -->Dual Norms</h1> | |||
Given any <a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space">Tp norm</a> on an interval space <strong>Tp<span style="font-size: 10px; vertical-align: super;">G</span></strong> associated with a group <strong>G</strong>, we can define a corresponding <strong>dual Tq* norm</strong> on the dual space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* which satisfies the following identity:<br /> | Given any <a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space">Tp norm</a> on an interval space <strong>Tp<span style="font-size: 10px; vertical-align: super;">G</span></strong> associated with a group <strong>G</strong>, we can define a corresponding <strong>dual Tq* norm</strong> on the dual space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* which satisfies the following identity:<br /> | ||
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