I have a problem where i need to show that $F = \mathcal{H}$ where $\mathcal{H}$ is a Hilbertspace and $F$ is a closed subspace of $\mathcal{H}$. I have worked out through the other information that $F$ is dense in $\mathcal{H}$ and from that i should be able to conclude that $F = \mathcal{H}$, but i don't really understand what it means that two spaces equal each other.
Thanks for reading
I think that the exercise is to show that $F$ is a dense and closed subspace in $H$ and conclude that $F$ must coincide with $H$ because $H$ is complete (i.e. there are no closed dense subspaces in $H$ except $H$ itself). It is a standard technique.