Let $H$ be a Hilbert space. Is it true that if $h_n \in H$ is a sequence such that $\sum ||h_n||^2 < \infty$, then $\sum h_n$ exists? I believe that this is false, but I cannot construct a counterexample. Obviously such sequence must not be orthogonal or even a Riesz-sequence.
2026-03-30 04:23:24.1774844604
Square summable sequence in Hilbert space is summable?
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No need to be to complicated ! $h_n=\frac{1}{n}$ and $H=\mathbb R$ will provide a (simple) counter-example.