Showing $P(\mathcal{H})$ is closed

Question

Suppose $\mathcal{H}$ is a Hilbert space and $\{u_i\}_{i=0}^\infty$ is an orthonormal family. Define $$P(w) = \sum_{i=0}^\infty \langle w, u_i \rangle u_i.$$ I wish to show that $P(\mathcal{H})$ is a closed subspace of $\mathcal{H}$. I can prove that it is a subspace, but not that it is closed. How would I do so?

Answer 1

Let $\mathcal{H}_1= $ the closed linear span of $\{u_i:i \in \mathbb N\}$.

Its your turn to show: $P(\mathcal{H})=\mathcal{H}_1$

Answer 2

$P$ is defined by a norm-convergent expansion and recognised as self-adjoint and idempotent, that is $P^*=P=P^2\,$. It is thus an orthogonal projector, by definition corresponding to the subspace spanned by the $u_i$.

And for a projection one generally has $$\operatorname{Im}P = \operatorname{Ker}(I-P)\,,$$ whence the claim.