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 $A$?
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. 

