Functional Analysis_16. The Riesz Representation Theorem_16.2. Series in Hilbert spaces

Let $X$ be a Hilbert space and $\{x_n\}_{n\in\mathbb{N}} \subset X$ an orthogonal system, i.e., \\ $\langle x_i, x_j \rangle = 0$ for all $i \neq j$. \pause Show that the following properties are equivalent: \begin{enumerate} \item[(i)] $\ds \sum_{n=1}^\infty x_n$ converges.\pause \item[(ii)] $\ds \sum_{n=1}^\infty \|x_n\|^2$ converges.\pause \item[(iii)] $\ds \sum_{n=1}^\infty x_n$ converges weakly. \end{enumerate}\pause (As in normed vector spaces, a series is called (weakly) convergent in a Hilbert space $X$ if the corresponding sequence of partial sums converges (weakly) in $X$.)