I'd like to prove the following:
If $\sum_{n=1}^\infty u_n = u$, then $\sum_{n=1}^\infty \langle u_n, x \rangle = \langle u, x \rangle$ for any $x$ in an inner product space.
Is the following proof valid?
$\sum_{n=1}^\infty u_n = u$ means that $\|\sum_{n=1}^N u_n - u\| \to 0$ as $N \to \infty$.
So, $|\sum_{n=1}^N \langle u_n, x \rangle - \langle u, x \rangle |$
$= | \langle \sum_{n=1}^N u_n - u, x \rangle |$, by linearity of inner products
$\le \| \sum_{n=1}^N u_n - u \| \| x \|$, by the Schwarz inequality
$\to 0$ as $N \to \infty$.
Therefore $\sum_{n=1}^\infty \langle u_n, x \rangle = \langle u, x \rangle$