In an inner product space, can we have $\langle\sum_{n=1}^\infty x_n,y\rangle=\sum_{n=1}^\infty\langle x_n,y\rangle$?

Question

In an inner product space $(X,\langle\cdot,\cdot\rangle)$, we have $\langle x_1+x_2,y\rangle=\langle x_1,y\rangle+\langle x_2,y\rangle$. If $\sum_{n=1}^\infty x_n$ is convergent in $X$, then can we have $\langle\sum_{n=1}^\infty x_n,y\rangle=\sum_{n=1}^\infty\langle x_n,y\rangle$?

Edit: If yes, how to prove it?