# On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces

On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces Let B be a separable Banach space and let X=B * be separable. We prove that if B has finite Szlenk index (for all ε > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if ε > 0 there exists δ (ε) > 0 so that if x n is a sequence in the ball of X converging ω* to x so that $$\lim \inf _{n \to \infty } \left\| {x_n - x} \right\| \geqslant \varepsilon {\text{ then }}\left\| x \right\| \leqslant 1 - \delta (\varepsilon )$$ . In addition we show that the norm can be chosen so that δ (ε) ≥ cεp for some p < ∞ and c >0. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces

Positivity, Volume 3 (2) – Sep 28, 2004
27 pages

/lp/springer_journal/on-asymptotic-structure-the-szlenk-index-and-ukk-properties-in-banach-DedPIk10mg
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009786603119
Publisher site
See Article on Publisher Site

### Abstract

Let B be a separable Banach space and let X=B * be separable. We prove that if B has finite Szlenk index (for all ε > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if ε > 0 there exists δ (ε) > 0 so that if x n is a sequence in the ball of X converging ω* to x so that $$\lim \inf _{n \to \infty } \left\| {x_n - x} \right\| \geqslant \varepsilon {\text{ then }}\left\| x \right\| \leqslant 1 - \delta (\varepsilon )$$ . In addition we show that the norm can be chosen so that δ (ε) ≥ cεp for some p < ∞ and c >0.

### Journal

PositivitySpringer Journals

Published: Sep 28, 2004

### References

• Property (M),M-ideals, and almost isometric structure of Banach spaces
Kalton, N. J.; Werner, D.

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