Does there exist any open linear (vector) subspace of a Hilbert space? I could not think of any example.
Actually, I was reading the book by Simmons, there almost in every theorem it assumed that "If M is a closed linear subspace".It seemed natural to me to think about subspaces which are not closed. I have an got an example which is not closed:
Take the Hilbert space H = L^[0,1], with L^2 norm and the subspace set of all polynomials, it is not closed because it's closure is H and not open can be found here Set of all polynomials on [0, 1/2] is not open in C[0, 1/2]. Then I asked myself an example of to think of an open set. But I could lead myself nowhere, as I am not familiar with infinite dimensional vector space. Not closed does not necessarily mean open.
Best Answer
If $M \leq \mathcal{H}$ a subspace of a Hilbert space (or generally any normed space) is open, then it contains a ball around the origin $0 \in B_r(0) \subset M$, but for every (none-zero) vector $v \in \mathcal{H}$, we have $$ \frac{r}{2\Vert v \Vert} v \in B_r(0) \subset M $$ But M is a linear subspace so $ v \in M $. Thus the only open subspaces of $ \mathcal{H} $ are $ \mathcal{H} $ itself.