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5
votes
Accepted
On a property for normed spaces
The separable spaces that do not have your property for bounded $(x_n)$ (namely, there is a sequence in the unit sphere satisfying your limit condition) are characterised in the paper ``Thickness of t …
2
votes
Accepted
Finding weak LUR property of $C[0,1]$ with an equivalent norm
Your norm, call it $N$ for \TeX-nical simplicity, is not WLUR. You want to know whether $N(f)=1$, $N(f_n)\to1$, $N(f+f_n)\to 2$ imply that $f_n\to f$ weakly. For a counterexample let $f$ be the const …