Generalized Tenney dual norms and Tp tuning space: Difference between revisions

Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space|Tp norm]] on an interval space '''Tp^G''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq^G'''* which satisfies the following identity:

For '''G''' with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm. Likewise, T1 interval space is dual to Tinf* tuning space, and T2 interval space is dual to T2* tuning space.

For an arbitrary group '''G''' with its chosen basis containing intervals other than primes and prime powers, || · ||'''_Tp''' is given by

Note that this is the quotient norm induced on the space '''Tq^L'''*/ker('''V_G'''), where ker('''V_G''') 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^L''' and our subspace is '''Tp^G''', this proves that our dual space '''Tq^G'''* must be isometrically isomorphic to '''Tq^L'''*/ker('''V_G''').