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If we have a Hilbert space $X$ and a closed subspace $M \subset X$, then using the projection theorem we can canonically find $M^\perp$ such that $X = M \oplus M^\perp$.

I'm wondering if it's possible to do an analogue to this in any topological vector space. Specifically, given a space $X$ and a closed $M \subset X$, can we decompose $X$ into $M \oplus \tilde{M}$ for some $\tilde{M}$ in a canonical way? Maybe this is a dumb question but it's not immediately clear to me that we can even complete a basis of $M$ to a basis of $X$.

whatev990
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    Briefly "no" even if $X$ is finite-dimensional. – Andrew D. Hwang Feb 21 '22 at 22:11
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    No. There are banach spaces with closed, noncomplemented subspaces. See here, say. In fact, the answer is "no" in a much deeper sense: If $X$ is a banach space where every closed subspace has an orthogonal complement, then actually $X$ is hilbert. See here. – Chris Grossack Feb 21 '22 at 22:11

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