Vol. 15, No. 3, 1965

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On the strict and uniform convexity of certain Banach spaces

Kondagunta Sundaresan

Vol. 15 (1965), No. 3, 1083–1086
Abstract

Let (X,S,μ) be a σ-finite non-atomic measure space let N be a real valued continuous convex even function defined on the real line such that

(1) N(u) is nondecreasing for u 0,

(2) limu→∞N(u)∕u = ,

(3) limu0N(u)∕u = 0.

Let LN be the set of all real valued μ-measurable functions f such that XN(f)dμ < . It is known that if there exists a constant k such that N(2u) kN(u) for all u 0 then LN is a linear space; in fact, LN is a B-Space if a norm ∥⋅∥ is defined by setting

                   ∫
∥f∥ = inf{1∕ζ | ζ > 0, N(η,f)dμ ≦ 1}.
X
(*)

Denoting the B-space (LN,∥⋅∥) by LN it is proposed to obtain the necessary and sufficient conditions in order that LN may be (1) Strictly Convex (2) Uniformly Convex.

Mathematical Subject Classification
Primary: 46.10
Milestones
Received: 20 January 1965
Published: 1 September 1965
Authors
Kondagunta Sundaresan