Sufficient conditions for uniform normal structure of Banach spaces and their duals

Convexity conditions and normal structure of Banach spaces

(b) Show that if X is uniformly convex and (xn) ⊂ X such that lim n→∞ ‖xn‖ = 1 and lim n,m→∞ ‖xn + xm‖ = 2, then (xn) is convergent. Proceed as follows. Use the definition to prove first that if all terms in the sequence have unit length, then the sequence is Cauchy. In the general case, let yn = xn/‖xn‖. Prove that lim n,m→∞ ‖yn + ym‖ = 2. Then, combine this with what you have shown already to prove that (xn) must be Cauchy. (c) Prove that if X is uniformly convex and S ⊂ X is closed and convex, then S contains a unique element of minimal norm, that is, a unique s0 ∈ S such that inf s∈S ‖s‖ = ‖s0‖. Proceed as follows. Let d = inf s∈S ‖s‖. Now, use part (b) to construct a

On Uniformly Convex Spaces

Let X be a Banach space and let S(X) ={ x ∈ X, � x �= 1}be the unit sphere of X. Parameters E(X) = sup{α(x), x ∈ S(X)}, e(X) = inf {α(x), x ∈ S(X)}, F(X) = sup{β(x), x ∈ S(X)} ,a ndf (X) = inf {β(x), x ∈ S(X)} ,w hereα(x) = sup{� x + y� 2 + � x − y� 2 , y ∈ S(X)} ,a ndβ(x) = inf {� x+ y� 2 + � x − y� 2 , y ∈ S(X)} are introduced and studied. The values of these parameters in the lp spaces and function spaces Lp[0,1] are estimated. Among the other results, we proved that a Banach space X with E(X) 2i s uniform nonsquare; and a Banach space X with E(X) 32/9 has uniform normal structure.

/pdf/a-pythagorean-approach-in-banach-spaces-2145u3ffay.pdf

A pythagorean approach in Banach spaces

Let X be a Banach space and be the unit sphere of X. Four new parameters , ; , , ,and , where , , ,, , , , are introduced and studied. The main result is that a Banach space X with has uniform normal structure.

Normal Structure, Fixed Points and Related Parameters in Banach Spaces

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Normal Structure, Fixed Points and Related Parameters in Banach Spaces