# does linearity of inner product hold for infinite sum?

this might be a dumb question but is the following true? $$<\sum_{n=1}^\infty x_n, y>=\sum_{n=1}^\infty <x_n, y>$$

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In a vector space, infinite sums do not make sense. If your vector space has a topology on it, and if your inner product is continuous with respect to that topology, then the answer is: yes. –  Mariano Suárez-Alvarez Feb 28 '11 at 2:23
This does not make sense because $n$ is free on the left hand side, but not on the right hand side. –  lhf Feb 28 '11 at 2:24
sorry im still not understanding...why dont direct sums make sense? –  jack Feb 28 '11 at 2:24
@lhf, fixed it. –  jack Feb 28 '11 at 2:26
@jack: Here's a small LaTeX tip: Use \langle ($\langle$) and \rangle ($\rangle$) instead of $\lt$ and $\gt$ when writing inner products. It looks better plus they can be autosized with amsmath's \left and \right commands: $$\left\langle \sum_{n=1}^\infty x_n, y_n \right\rangle.$$ Right click the math above to see the source –  kahen Feb 28 '11 at 2:47
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$$\langle\sum_{n=1}^\infty x_n,y\rangle=\langle\lim_{N\rightarrow\infty}\sum_{n=1}^N x_n, y\rangle=\lim_{N\rightarrow\infty}\langle\sum_{n=1}^N x_n,y\rangle=\lim_{N\rightarrow\infty}\sum_{n=1}^N\langle x_n,y\rangle=\sum_{n=1}^\infty \langle x_n,y\rangle$$