Question

What is physical interpretation of a closed subspace of a separable hilbert space?

Answer 1

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. 
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Answer 2

Explanation:

Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. The origin of the designation "der abstrakte Hilbertsche Raum" is John von Neumann in his famous work on unbounded Hermitian operators published in 1929.