Is the pythagoras theorem true for an infinite set of vectors?

Question

Let $H$ be a Hilbert space. Consider a collection of orthogonal vectors $\{x_n\}_{n=1}^{k}$. We know that $$\left\|\sum_{n=1}^{k}x_n\right\|^2=\sum_{n=1}^{k}\|x_n\|^2$$

Now consider an infinite collection of orthogonal vectors $\{y_n\}_{n=1}^{\infty}.$ Is it true that $$\left\|\sum_{n=1}^{\infty}y_n\right\|^2=\sum_{n=1}^{\infty}\|y_n\|^2$$

Answer 1

\begin{align} \left\|\sum_{n=1}^{\infty}y_n\right\|^2 &=\left\langle\sum_{n=1}^{\infty}y_n,\sum_{k=1}^{\infty}y_k\right\rangle\\&=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\left\langle y_n,y_k\right\rangle\\&=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\left\langle y_n,y_k\right\rangle\delta_{k,n}\\&=\sum_{n=1}^{\infty}\left\langle y_n,y_n\right\rangle\\&=\sum_{n=1}^{\infty}\|y_n\|^2. \end{align}