CHARACTERISTIC INEQUALITIES OF UNIFORMLY CONVEX AND UNIFORMLY SMOOTH BANACH-SPACES

Let X be a real Banach space with dual X* and moduli of convexity and smoothness δX(ε) and ρ{variant}X(τ), respectively. For 1 < p <∞, Jp denotes the duality mapping from X into 2X with gauge function tp - 1 and jp denotes an arbitrary selection for Jp. Let A= {φ: R+ → + : φ (0) = 0, φ(t) is strictly increasing and there exists c > 0 such that φ(t) ≥ cδX( t 2)} and F= {θ{symbol}: R+ → + : θ{symbol} (0) = 0, θ{symbol}(t) is convex, nondecreasing and there exists K > 0 such that θ{symbol}(τ) ≤ Kρ{variant}X(τ)}. It is proved that X is uniformly convex if and only if there is a φ ε{lunate} A such that ∥x + y∥p ≥ ∥x∥p + p〈jpx, y〉 + σφ(x, y) ∀x, y ε{lunate} X and X is uniformly smooth if and only if there is a θ{symbol} ∈ F such that ∥x + y∥p ≤ ∥x∥p + p〈jpx, y〉 + σθ{symbol}(x, y) ∀x, y ε{lunate} X, where, for given function f, σf(x, y) is defined by σ f (x,y) = p∫ 01 ∥ x+ ty ∥ ∨ ∥ x ∥) p t f t ∥ y ∥ ∥ x+ty ∥ ∨ ∥ x ∥dt These inequalities which have various applications can be regarded as general Banach space versions of the well-known polarization identity occurring in Hilbert spaces. © 1970.