NDLI: Power-normed spaces

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Power-normed spaces

Content Provider	Springer Nature Link
Author	Blasco, O.
Copyright Year	2016
Abstract	To each power-norm $$((E^n, \Vert \cdot \Vert _n):n\in {\mathbb N})$$ based on a given Banach space E, we associate two maximal symmetric sequence spaces $$L_\Phi ^E$$ and $$L_\Psi ^E$$ whose norms $$\Vert (z_k)\Vert _{L_\Phi ^E}$$ and $$\Vert (z_k)\Vert _{L_\Psi ^E}$$ are defined by $$\sup \{ \Vert (z_1x,\ldots ,z_nx)\Vert _n: \Vert x\Vert =1, n\in {\mathbb N}\}$$ and $$\sup \{ \Vert \sum _{k=1}^n z_kx_k\Vert : \Vert (x_1,\ldots ,x_n)\Vert _n=1, n\in {\mathbb N}\}$$ respectively. For each $$1\le p\le \infty $$ , we introduce and study the p-power-norms as those power-norms for which $$L_\Phi ^E=\ell ^p$$ and $$L_\Psi ^E=\ell ^{p'}$$ , where $$1/p+1/p'=1$$ . As a special cases of p-power-norms we introduce certain smaller class, to be called the class of $$\ell ^p$$ -power-norms, which is shown to contain the p-multi-norms defined in (Dales et al., Multi-norms and Banach lattices, 2016), and to coincide with the multi-norms and dual-multi-norms defined in (Dales and Polyakov, Diss Math 488, 2012) in the cases $$p=\infty $$ and $$p=1$$ respectively. We give several procedures to construct examples of such p-power and $$\ell ^p$$ -power-norms and show that the natural formulations of the (p, q)-summing, (p, q)-concave, Rademacher power norms, t-standard power norms among others are examples in these classes. In particular, for instance the Rademacher power norm is a 2-power norm and the (p, q)-summing power-norm is a $$\ell ^r$$ -power-norm for $$p>q$$ with $$\frac{1}{r}=\frac{1}{q}-\frac{1}{p}$$ .
Ending Page	632
Page Count	40
Starting Page	593
File Format	PDF
ISSN	13851292
e-ISSN	15729281
Journal	Positivity
Issue Number	2
Volume Number	21
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-03-19
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Multinorms Interpolation between normed linear spaces p-multinorms Geometry and structure of normed linear spaces Power-norms Rademacher norm Banach lattices (p, q)-concave norm Isomorphic theory (including renorming) of Banach spaces (p, q)-summing norm Operator Theory Fourier Analysis Potential Theory Calculus of Variations and Optimal Control; Optimization Econometrics
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Analysis

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