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A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible subspace for a subalgebra $A \in B(\mathcal{H})$. Is it any subspace that does not contain a proper invariant subspace under $A$, or a closed subspace that does not contain a proper invariant subspace under $A$?

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Typically, one only considers closed subspaces of Hilbert spaces.

If someone ever were to consider not necessarily closed subspaces, I would expect them to say that explicitly.

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In Halmos, he uses "subspace" if it is a closed linear subspace, and "linear manifold" if it is a linear subspace, not necessarily closed. The linear manifold is rarely used in the study of bounded linear operators. – Gerald Edgar Jun 13 '12 at 12:39
Small addedndum to Gerald Edgar's remarks: there has been some work on "operator ranges" (also called "paraclosed subspaces"), namely the linear manifolds inside $H$ that are of the form $T(H)$ for some continuous linear operator $T:H\to H$. – Yemon Choi Jun 13 '12 at 22:51

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