# Power-normed spaces

Power-normed spaces To each power-norm $$((E^n, \Vert \cdot \Vert _n):n\in {\mathbb N})$$ ( ( E n , ‖ · ‖ n ) : n ∈ N ) based on a given Banach space E, we associate two maximal symmetric sequence spaces $$L_\Phi ^E$$ L Φ E and $$L_\Psi ^E$$ L Ψ E whose norms $$\Vert (z_k)\Vert _{L_\Phi ^E}$$ ‖ ( z k ) ‖ L Φ E and $$\Vert (z_k)\Vert _{L_\Psi ^E}$$ ‖ ( z k ) ‖ L Ψ E are defined by $$\sup \{ \Vert (z_1x,\ldots ,z_nx)\Vert _n: \Vert x\Vert =1, n\in {\mathbb N}\}$$ sup { ‖ ( z 1 x , … , z n x ) ‖ n : ‖ x ‖ = 1 , n ∈ N } and $$\sup \{ \Vert \sum _{k=1}^n z_kx_k\Vert : \Vert (x_1,\ldots ,x_n)\Vert _n=1, n\in {\mathbb N}\}$$ sup { ‖ ∑ k = 1 n z k x k ‖ : ‖ ( x 1 , … , x n ) ‖ n = 1 , n ∈ N } respectively. For each $$1\le p\le \infty$$ 1 ≤ p ≤ ∞ , we introduce and study the p-power-norms as those power-norms for which $$L_\Phi ^E=\ell ^p$$ L Φ E = ℓ p and $$L_\Psi ^E=\ell ^{p'}$$ L Ψ E = ℓ p ′ , where $$1/p+1/p'=1$$ 1 / p + 1 / p ′ = 1 . As a special cases of p-power-norms we introduce certain smaller class, to be called the class of $$\ell ^p$$ ℓ 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 $$p=\infty$$ p = ∞ and $$p=1$$ p = 1 respectively. We give several procedures to construct examples of such p-power and $$\ell ^p$$ ℓ 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 $$\ell ^r$$ ℓ r -power-norm for $$p>q$$ p > q with $$\frac{1}{r}=\frac{1}{q}-\frac{1}{p}$$ 1 r = 1 q - 1 p . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Power-normed spaces

, Volume 21 (2) – Mar 19, 2016
40 pages

/lp/springer_journal/power-normed-spaces-yc5aZMG3DW
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0404-6
Publisher site
See Article on Publisher Site

### Abstract

To each power-norm $$((E^n, \Vert \cdot \Vert _n):n\in {\mathbb N})$$ ( ( E n , ‖ · ‖ n ) : n ∈ N ) based on a given Banach space E, we associate two maximal symmetric sequence spaces $$L_\Phi ^E$$ L Φ E and $$L_\Psi ^E$$ L Ψ E whose norms $$\Vert (z_k)\Vert _{L_\Phi ^E}$$ ‖ ( z k ) ‖ L Φ E and $$\Vert (z_k)\Vert _{L_\Psi ^E}$$ ‖ ( z k ) ‖ L Ψ E are defined by $$\sup \{ \Vert (z_1x,\ldots ,z_nx)\Vert _n: \Vert x\Vert =1, n\in {\mathbb N}\}$$ sup { ‖ ( z 1 x , … , z n x ) ‖ n : ‖ x ‖ = 1 , n ∈ N } and $$\sup \{ \Vert \sum _{k=1}^n z_kx_k\Vert : \Vert (x_1,\ldots ,x_n)\Vert _n=1, n\in {\mathbb N}\}$$ sup { ‖ ∑ k = 1 n z k x k ‖ : ‖ ( x 1 , … , x n ) ‖ n = 1 , n ∈ N } respectively. For each $$1\le p\le \infty$$ 1 ≤ p ≤ ∞ , we introduce and study the p-power-norms as those power-norms for which $$L_\Phi ^E=\ell ^p$$ L Φ E = ℓ p and $$L_\Psi ^E=\ell ^{p'}$$ L Ψ E = ℓ p ′ , where $$1/p+1/p'=1$$ 1 / p + 1 / p ′ = 1 . As a special cases of p-power-norms we introduce certain smaller class, to be called the class of $$\ell ^p$$ ℓ 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 $$p=\infty$$ p = ∞ and $$p=1$$ p = 1 respectively. We give several procedures to construct examples of such p-power and $$\ell ^p$$ ℓ 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 $$\ell ^r$$ ℓ r -power-norm for $$p>q$$ p > q with $$\frac{1}{r}=\frac{1}{q}-\frac{1}{p}$$ 1 r = 1 q - 1 p .

### Journal

PositivitySpringer Journals

Published: Mar 19, 2016

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