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.
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Knaust, H., Odell, E. & Schlumprecht, T. On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces. Positivity 3, 173–200 (1999). https://doi.org/10.1023/A:1009786603119
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DOI: https://doi.org/10.1023/A:1009786603119