Can anyone tell me the physical interpretation of a closed subspace of separable hilbert space? I would like to know since quantum mechanics is best described in hilbert space and we have a physical interpretation of hilbert space in context to a physical system. Please keep in mind the context of the invariant subspace problem (whether every bounded linear operator on a separable hilbert space has non trivial invariant closed subspace or not).

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    $\begingroup$ Closed sub spaces are important because they are related to projection operators. The values of observable are given by eigenvalues of hermitian operators. Each of these events will be represented by projection onto the corresponding eigenspace. $\endgroup$ – Boltzee Jun 8 '17 at 16:34

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