Generalized Tenney dual norms and Tp tuning space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 356537958 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 356537966 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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 12:16: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 12:16:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>356537966</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> | ||
| Line 10: | Line 10: | ||
[[math]] | [[math]] | ||
||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \} | ||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \} | ||
[[math]] | [[math]] | ||
| Line 19: | Line 19: | ||
[[math]] | [[math]] | ||
\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p} | \left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p} | ||
[[math]] | [[math]] | ||
| Line 25: | Line 25: | ||
[[math]] | [[math]] | ||
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q} | \left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q} | ||
[[math]] | [[math]] | ||
| Line 38: | Line 38: | ||
[[math]] | [[math]] | ||
\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1} | \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1} | ||
[[math]] | [[math]] | ||
| Line 44: | Line 44: | ||
[[math]] | [[math]] | ||
\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 \} | \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 \} | ||
[[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 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><!-- 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> | ||
| Line 54: | Line 54: | ||
<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}&lt;br/&gt;[[math]] | ||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex"> ||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | <br /> | ||
for all f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>*. This normed space, for which the group of vals on <strong>G</strong> comprise the lattice of covectors with integer coefficients, is called <strong>Tq* Tuning Space.</strong><br /> | for all f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>*. This normed space, for which the group of vals on <strong>G</strong> comprise the lattice of covectors with integer coefficients, is called <strong>Tq* Tuning Space.</strong><br /> | ||
| Line 64: | Line 64: | ||
<!-- ws:start:WikiTextMathRule:1: | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}&lt;br/&gt;[[math]] | \left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex"> \left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
<br /> | <br /> | ||
for diagonal weighting matrix <strong>W<span style="font-size: 80%; vertical-align: sub;">G</span></strong>. Then the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq*</span></strong> on <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* is given for f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* by<br /> | for diagonal weighting matrix <strong>W<span style="font-size: 80%; vertical-align: sub;">G</span></strong>. Then the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq*</span></strong> on <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* is given for f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* by<br /> | ||
| Line 71: | Line 71: | ||
<!-- ws:start:WikiTextMathRule:2: | <!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}&lt;br/&gt;[[math]] | \left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex"> \left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
<br /> | <br /> | ||
where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.<br /> | where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.<br /> | ||
| Line 85: | Line 85: | ||
<!-- ws:start:WikiTextMathRule:3: | <!-- ws:start:WikiTextMathRule:3: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]] | \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | --><script type="math/tex"> \left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
<br /> | <br /> | ||
for a <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></strong> representing <strong>G</strong> in some full-limit <strong>L</strong> and a diagonal weighting matrix <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> for <strong>L</strong>. Then if <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> represents the full-limit interval space that <strong>G</strong> is embedded in, and <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>* is the dual space, the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq </span></strong>on <strong>Tq<span style="font-size: 80%; vertical-align: super;">G</span></strong>* is given by<br /> | for a <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></strong> representing <strong>G</strong> in some full-limit <strong>L</strong> and a diagonal weighting matrix <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> for <strong>L</strong>. Then if <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> represents the full-limit interval space that <strong>G</strong> is embedded in, and <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>* is the dual space, the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq </span></strong>on <strong>Tq<span style="font-size: 80%; vertical-align: super;">G</span></strong>* is given by<br /> | ||
| Line 92: | Line 92: | ||
<!-- ws:start:WikiTextMathRule:4: | <!-- ws:start:WikiTextMathRule:4: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\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 \}&lt;br/&gt;[[math]] | \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 \}&lt;br/&gt;[[math]] | ||
--><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 /> | ||
<br /> | <br /> | ||
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> | 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> | ||