Abstract
Inspired by the concept of U‐spaces introduced by Lau, (1978), we introduced the class of semi‐uniform Kadec‐Klee spaces, which is a uniform version of semi‐Kadec‐Klee spaces studied by Vlasov, (1972). This class of spaces is a wider subclass of spaces with weak normal structure and hence generalizes many known results in the literature. We give a characterization for a certain direct sum of Banach spaces to be semi‐uniform Kadec‐Klee and use this result to construct a semi‐uniform Kadec‐Klee space which is not uniform Kadec‐Klee. At the end of the paper, we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al., (1997).
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