Vtyjkyuk

Let $H$ be a Hilbert space and $Ssubseteq H$ a closed subspace. Moreover, let $s_n_n=1^inftysubseteq S$ a complete and linear independent sequence in $S$, i.e.

• $S=overlinetextSpanbig(s_n_n=1^inftybig)$ and


• for $Ngeq 1$ and $lambdainmathbbR^N$, $sum_n=1^Nlambda_n,s_n=0$ implies $lambda=0$.

Denote by $mathcalP$ the orthogonal projection from $H$ onto $S$ and by $mathcalP_N$ the orthogonal projection from $H$ onto $textSpanbig(s_n_n=1^Nbig)$.

Is it true that, for all $xin H$,
$$mathcalP_N(x) quadrightarrowquad mathcalP(x)$$ for
$Nrightarrowinfty$ ?

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.

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 $operatornamespan e_1, ldots, e_n = operatornamespan s_1, ldots, s_n, forall n in mathbbN$ and $operatornamespan e_n_n=1^infty = operatornamespan s_n_n=1^infty$.

Then clearly $e_n_n=1^infty$ is an orthonormal basis for $S$ so

$$P_nx = sum_k=1^n langle x, e_krangle e_k xrightarrowntoinfty sum_k=1^infty langle x, e_krangle e_k = Px$$

for all $x in H$.