I have a question about a statement which I don't know if it is true. Given a Hilbert space $H$, if we have a sequence of element $\{c_k\}$, with $||\sum_I c_k||\leq A$, where $I$ is any finite index set and $A$ is independent of $I$. Can we say $\sum||c_k||^2<\infty$? Thanks for any hint!
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if $K$ is compact then every sequence in $K$ has a convergent subsequence. – reuns Aug 25 '16 at 00:41
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Thanks for your comment! Can you explain it in more detail? Thanks！ – Ale Aug 26 '16 at 02:00