Is there any space with normal structure but not uniform normal structure?

Question

It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some example? thanks...

Answer 1

In example 4.1 of E. Maluta, Uniformly normal structure and related coefficients, Pac. J. Math, 111 (1984), No. 2, 357–369, it is shown that the $\ell_2$-sum $Y = \bigoplus_{n \geq 2} \ell_n$ is reflexive, has normal structure, but not uniformly normal structure.

In fact, Maluta investigates two numerical characteristics $D(X)$ and $\tilde{N}(X)$ of a Banach space $X$. The space $X$ has uniformly normal structure iff $\tilde{N}(X) \lt 1$. Her Theorem 4.2 shows that $D(X) \leq \tilde{N}(X)$ for all Banach spaces, and in her Example 4.1 she shows that $D(Y) = 1$ for the example above.

The main result of the paper is that a space having uniformly normal structure is necessarily reflexive.