Fourier Coefficients in arbitrary Hilbert Spaces

Say we have an orthonormal basis $\{e_n\}$ for a infinite Hilbert Space $H$. I want to prove that any vector $x=\sum_{n=1}^\infty\langle x, e_n\rangle e_n$. I don't know where to start. Could I have any help?

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 I took the liberty of correcting "$$" to "\langle x,e_n\rangle e_n". Clearly x isn't a sum of scalars unless H = \mathbb K. – kahen Sep 1 '12 at 1:59 Thank you. My mistake. I didn't read over this after I posted. – Parakee Sep 1 '12 at 2:02 2 Answers What is the meaning of$$ x=\sum_{n=1}^\infty \langle x,e_n\rangle e_n? $$This is$$ x=\lim_{m\to\infty}\sum_{n=1}^m \langle x,e_n\rangle e_n, $$or in other words$$ \left\|x-\sum_{n=1}^m \langle x,e_n\rangle e_n\right\|\to 0,  as $m\to\infty$. Now recall $\|y\|^2=\langle y,y\rangle$, and try to show that the preceding limit goes to $0$.

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Consider the set of all vectors of that form. Prove that it is a (closed) subspace, and that its orthogonal is zero. Then it has to be all of $H$.

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