does linearity of inner product hold for infinite sum?

Question

This might be a dumb question but is the following true? $$\left \langle\sum_{n=1}^\infty x_n, y\right \rangle=\sum_{n=1}^\infty \langle x_n, y \rangle$$

Answer 1

As Mariano points out, you need to have a topological vector space - otherwise there is no such thing as an infinite sum, because an "infinite sum" is really defined to be a limit of partial sums, and no topology means no limits. However, if there is a topology, and the inner product is continuous with respect to it, we have

$$\langle\sum_{n=1}^\infty x_n,y\rangle=\langle\lim_{N\rightarrow\infty}\sum_{n=1}^N x_n, y\rangle=\lim_{N\rightarrow\infty}\langle\sum_{n=1}^N x_n,y\rangle=\lim_{N\rightarrow\infty}\sum_{n=1}^N\langle x_n,y\rangle=\sum_{n=1}^\infty \langle x_n,y\rangle$$

where the second equality is justified by the assumption that the inner product is continuous and hence preserves limits, and the third equality is justified by the bilinearity of the inner product.