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Let $H$ be a Hilbert space, such that $H = W_1 \bigoplus W_2$, i.e., $H$ is direct sum of two subspaces $W_1$ and $W_2$.

Is it true that $W_1,W_2$ are closed ? If I assume the Axiom of Choice, I can show the existence of a counter example. But can someone give a solid counter example ?

Kr Dpk
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  • The approved answer here may be useful: https://math.stackexchange.com/questions/2689457/example-of-an-infinite-dimensional-hilbert-space-that-is-not-an-rkhs/2690389#2690389 – Kavi Rama Murthy Oct 19 '21 at 11:33
  • Is the direct sum an orthogonal direct sum? I.e. do you assume $W_1 \perp W_2?$ – J. De Ro Oct 19 '21 at 18:58
  • @QuantumSpace With that assumption, it can be proved. I am not making that assumption. – Kr Dpk Oct 19 '21 at 19:02

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