Generalized Tenney dual norms and Tp tuning space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 356537652 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 356537958 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12:16:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>356537958</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
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> | ||
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[[math]] | [[math]] | ||
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 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><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Dual Norms"></a><!-- ws:end:WikiTextHeadingRule:5 -->Dual Norms</h1> | <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> | ||
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--><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | --><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
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Note that this is the quotient norm induced on the space <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>), where ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the <a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow" target="_blank">Hahn-Banach theorem</a>, which demonstrates that the space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* must be isometrically isomorphic to <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>).</body></html></pre></div> | Note that this is the quotient norm induced on the space <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>), where ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the <a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow" target="_blank">Hahn-Banach theorem</a>, 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 <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> and our subspace is <strong>Tp<span style="font-size: 10px; vertical-align: super;">G</span></strong>, this proves that our dual space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* must be isometrically isomorphic to <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>).</body></html></pre></div> | ||