Generalized Tenney dual norms and Tp tuning space: Difference between revisions
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<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 \}</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 \}</math> | ||
Note that this is the quotient norm induced on the space '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>'''), where ker('''V<sub>G</sub>''') is the set of vals | Note that this is the quotient norm induced on the space '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>'''), where ker('''V<sub>G</sub>''') is the set of vals (or tuning maps) that are restricted away given the V_map. 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<sup>L</sup>''' and our subspace is '''Tp<sup>G</sup>''', this proves that our dual space '''Tq<sup>G</sup>'''* must be isometrically isomorphic to '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>'''). | ||
[[category:math]] | [[category:math]] | ||