Convergence of a series in a Hilbert space

Question

Let $\{y_k:k=1,2,\dots\}$ be an orthonormal set in a Hilbert space $\mathcal{H}$. Give and prove the necessary and sufficient conditions for the scalar sequence $\{\lambda_k\}$ such that the series $$\sum_{k=1}^\infty\lambda_k y_k$$ converges. What is the norm of the sum of the series then?

This is my preparatory task before an exam, but I'm struggling with the first step, could you explain in detail what I'm supposed to do and what theorems to make use of?

Answer 1

Hint: $\sum \lambda_k y_k$ converges $\iff$ $\sum |\lambda_k|^2$ converges.

Sketch of proof: since a Hilbert space is complete, the two series converge iff their partial sums are Cauchy. Since $(y_n)_{n\in \mathbb N}$ is orthonormal, by Pythagoras' Theorem the Cauchy conditions for the two partial sums are equivalent.

Edit: let $$S_n:=\sum_{k=1}^n \lambda_k y_k \qquad \sigma_n:=\sum_{k=1}^n|\lambda_k|^2.$$ Then, as a conseguence of Pythagoras' theorem, $$\begin{align} ||S_{n+p}-S_n||^2=& ||\sum_{k=n}^{n+p}\lambda_k y_k||^2 \\= & \sum_{k=n}^{n+p}|\lambda_k|^2||y_k||^2 \\= & \sum_{k=n}^{n+p}|\lambda_k|^2 \\= &\sigma_{n+p}-\sigma_n \end{align} $$ From here it is clear that $(S_n)_{n\in \mathbb N}$ is Cauchy iff $(\sigma_n)_{n\in \mathbb N}$ is Cauchy, which concludes the proof.

Answer 2

Hint: for $n \in \mathbb N$ we have

$||\sum_{k=1}^n\lambda_k y_k||^2=\sum_{k=1}^n |\lambda_k|^2 $.

Its now your turn to show:

$\sum_{k=1}^\infty\lambda_k y_k$ converges $ \iff (\lambda_k) \in l^2$.