Projection onto the span of linearly independent vectors in Hilbert spaces

Question

Let $H$ be a Hilbert space and $S\subseteq H$ a closed subspace. Moreover, let $\{s_{n}\}_{n=1}^{\infty}\subseteq S$ a complete and linear independent sequence in $S$, i.e.

• $S=\overline{\text{Span}\big(\{s_{n}\}_{n=1}^{\infty}\big)}$ and
• for $N\geq 1$ and $\lambda\in\mathbb{R}^{N}$, $\sum_{n=1}^{N}\lambda_{n}\,s_{n}=0$ implies $\lambda=0$.

Denote by $\mathcal{P}$ the orthogonal projection from $H$ onto $S$ and by $\mathcal{P}_{N}$ the orthogonal projection from $H$ onto $\text{Span}\big(\{s_{n}\}_{n=1}^{N}\big)$.

Is it true that, for all $x\in H$, $$\mathcal{P}_{N}(x) \quad\rightarrow\quad \mathcal{P}(x)$$ for $N\rightarrow\infty$ ?

This is used in a paper without proof or comment and I am wondering how to show this rigorously.

Any help or comment is highly appreciated! Thanks.

Answer 1

Apply Gram-Schmidt process on the sequence $\{s_n\}_{n=1}^\infty$ to obtain an orthonormal sequence $\{e_n\}_{n=1}^\infty$ in $S$ such that $\operatorname{span} \{e_1, \ldots, e_n\} = \operatorname{span} \{s_1, \ldots, s_n\}, \forall n \in \mathbb{N}$ and $\operatorname{span} \{e_n\}_{n=1}^\infty = \operatorname{span} \{s_n\}_{n=1}^\infty$.

Then clearly $\{e_n\}_{n=1}^\infty$ is an orthonormal basis for $S$ so

$$P_nx = \sum_{k=1}^n \langle x, e_k\rangle e_k \xrightarrow{n\to\infty} \sum_{k=1}^\infty \langle x, e_k\rangle e_k = Px$$

for all $x \in H$.