Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications
Prob. 8, Sec. 3.5
$\DeclareMathOperator{\span}{span}$Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$, and let $M = \span (e_k)$. Let $x \in H$.
If $$x = \sum_{k=1}^\infty \langle x, e_k \rangle e_k,$$ then $x \in \overline{\span(e_k)}$ because in this case the sequence $(s_n)$ in $\span(e_k)$, where $s_n = \sum_{k=1}^n \langle x, e_k \rangle e_k$, converges to $x$.
How to show the converse?
That is, how to show that if $x \in \overline{\span(e_k)}$, then the series $\sum_{k=1}^\infty \langle x, e_k \rangle e_k$ converges (in the norm induced by the inner product on $H$) and has sum $x$?
My effort:
Suppose $x \in \overline{\span(e_k)}$. Then there is a sequence $(x_n)$ in $\span(e_k)$ that converges to $x$. Let $x_n = \sum_{k=1}^{m_n} \alpha_{nk} e_k$ for each $n= 1, 2, 3, \ldots$, where $\alpha_{nk}$ are scalars and the $m_n$ are natural numbers.
Then, using the orthonormality of the $e_k$, we can conclude that $\alpha_{nk} = \langle x_n, e_k \rangle$ for each $n=1, 2, 3, \ldots$ and for each $k= 1, \ldots, m_n$. So $$x_n = \sum_{k=1}^{m_n} \langle x_n, e_k \rangle e_k. $$
What next?
Can we say the following?
For each fixed $k$, $$\langle x_n, e_k \rangle \to \langle x, e_k \rangle \ \mbox{ as } \ n \to \infty. $$
How to show that $$x = \sum_{k=1}^\infty \langle x, e_k \rangle e_k?$$
I also know that the series $\sum \langle x, e_k \rangle e_k$ does converge.
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