P-Uniform Convexity andq-Uniform Smoothness of Absolute Normalized Norms onℂ2

Tomonari Suzuki

doi:10.1155/2014/746309

Abstract

We first prove characterizations ofp-uniform convexity andq-uniform smoothness. We next give a formulation on absolute normalized norms onℂ2. Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is notp-uniformly convex.

Highlights

We present some examples of Banach spaces
Throughout this paper, we denote by N, R, and C the sets of positive integers, real numbers, and complex numbers, respectively
We let Ψ2 be the set of all convex functions ψ on [0, 1] satisfying max {1 − t, t} ≤ ψ (t) ≤ 1 (7)

Summary

Introduction

Throughout this paper, we denote by N, R, and C the sets of positive integers, real numbers, and complex numbers, respectively. For p ∈ [2, ∞), X is called p-uniformly convex if there exists C > 0 satisfying δ (ε) ≥ Cεp (3). For p ∈ (1, ∞), Lp spaces are max{2, p}-uniformly convex and min{2, p}-uniformly smooth. Let AN2 be the family of all absolute normalized norms on C2. We let Ψ2 be the set of all convex functions ψ on [0, 1] satisfying max {1 − t, t} ≤ ψ (t) ≤ 1. Saito et al in [8] extended In this paper, we first prove characterizations of puniform convexity and q-uniform smoothness. We give another formulation on absolute normalized norms on C2. We present some examples, one of which is a uniformly convex Banach space which is not p-uniformly convex

Characterizations

Convex Functions

Absolute Normalized Norms on C2

Examples