Power-normed spaces

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis 2017-06, Vol.21 (2), p.593-632
• Main Author: Blasco, O.
• Format: Article
• Language: English
• Subjects: Banach spaces; Calculus of Variations and Optimal Control; Optimization; Econometrics; Formulations; Fourier Analysis; Lattices (mathematics); Mathematical analysis; Mathematics; Mathematics and Statistics; Norms; Operator Theory; Potential Theory
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-016-0404-6

To each power-norm ( ( E n , ‖ · ‖ n ) : n ∈ N ) based on a given Banach space E , we associate two maximal symmetric sequence spaces L Φ E and L Ψ E whose norms ‖ ( z k ) ‖ L Φ E and ‖ ( z k ) ‖ L Ψ E are defined by sup { ‖ ( z 1 x , … , z n x ) ‖ n : ‖ x ‖ = 1 , n ∈ N } and sup { ‖ ∑ k = 1 n z k x k ‖ : ‖ ( x 1 , … , x n ) ‖ n = 1 , n ∈ N } respectively. For each 1 ≤ p ≤ ∞ , we introduce and study the p -power-norms as those power-norms for which L Φ E = ℓ p and L Ψ E = ℓ 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 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 = ∞ and p = 1 respectively. We give several procedures to construct examples of such p -power and 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 ℓ r -power-norm for p > q with 1 r = 1 q - 1 p .