The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces

Abstract

Abstract. We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space 2,∞ are computed, and are used to show that our results are sharp.

On James and von Neumann–Jordan constants and sufficient conditions for the fixed point property

“…The assertion follows immediately from the fact that C NJ (X * ) = C NJ (X) and w(X * ) = w(X) [11,Theorem 5]. Finally, let us consider the space X = l 2,1 , which is l 2 renormed according to x 2,1 := x + 2 + x − 2 , where x + and x − are the positive and the negative part or x, respectively, defined as x + = (max{x(i), 0}) and [2]).…”

“…Finally, let us consider the space X = l 2,1 , which is l 2 renormed according to x 2,1 := x + 2 + x − 2 , where x + and x − are the positive and the negative part or x, respectively, defined as x + = (max{x(i), 0}) and [2]). It was proved in [11,Theorem 8] that C NJ (l 2,1 ) = 3/2 and it is easy to see that w(l 2,1 ) = 1/ √ 2, and R(l 2,1 ) = 2. 2…”

Constructing separated sequences in Banach spaces

“…Then e n = 1 = e n − e m for every n and m = n. Consequently, W CS(X) = 1. Results in [16] and [5] show that g(X) = 2 √ 2/(1+ √ 2) > W CS(X). We will show that s(X) = K(X) = C 1 (2, X) = √ 2 > g(X).…”

The characteristic of convexity of a Banach space and normal structure

“…We also note here that sometimes calculating the characteristic of convexity of the dual space is much easier than calculating the James or von Neumann-Jordan constants of the space (see [1,Example IV.7] and [10,Theorem 4]). …”