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 ?

  • $\begingroup$ The approved answer here may be useful: math.stackexchange.com/questions/2689457/… $\endgroup$ Oct 19, 2021 at 11:33
  • $\begingroup$ Is the direct sum an orthogonal direct sum? I.e. do you assume $W_1 \perp W_2?$ $\endgroup$
    – J. De Ro
    Oct 19, 2021 at 18:58
  • $\begingroup$ @QuantumSpace With that assumption, it can be proved. I am not making that assumption. $\endgroup$
    – Kr Dpk
    Oct 19, 2021 at 19:02


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