So, I am sitting waiting for my eye appointment, and idly reading Halmos on Hilbert spaces (strange looks from my neighbours!).

I said "idly" as the very early stuff was quite familiar, but then I get to this, which is almost (not quite) verbatim.

Suppose that

be an Hilbert space (usual definition), and that

is a subset.

will be a

*linear manifold* if, for all

and any

that

.

Fair enough, we have a definition. But now....

If

is a

*closed linear manifold* then it is a {vector} subspace of

.

OK, I know the general definition of a manifold, and I know what the closed subsets of a topological space are, but I have never come across the notion of a closed manifold, linear or otherwise.

And why does this make it a vector subspace? I am sure I am being dim here, but any help would be welcomed.

PS There are some obvious questions that arise from this construction: Like, does this imply that every Hilbert space is a manifold, closed, linear or otherwise? Some are quite trivially manifolds in the "naive" sense (like

and

for any

, for example), but is it generally true that all Hilbert spaces are closed linear manifolds, whatever that means?