Any example of non-closed operator?

Question

I cannot think of one.

By the way, is there any good exercise book on functional analysis or hilbert space?

Answer 1

Any bounded operator $A$ on a dense vector subspace $D$ of a Hilbert space $H$ such that $D\neq H.$

Answer 2

For a concrete example of an operator that isn't even closable, consider $$ M: (x_1,x_2,x_3,\ldots,x_n,\ldots)\mapsto(x_1,2x_2,3x_2,\ldots,nx_n,\ldots)$$ defined on the subspace of sequences with bounded support in $\ell^2$.

Then the sequence $(\frac 1n \mathbf e_n)_n$ clearly converges (to 0), but $(M(\frac 1n\mathbf e_n))_n = (\mathbf e_n)_n$ doesn't.