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Let $H_1, H_2, \ldots $ be a countable set of Hilbert spaces. Let $H\subset \prod_k H_k$ be the set where

$$\|x\|^2 = \sum_k \|x_k\|^2_{H_k} < \infty.$$

Show that $H$ is a Hilbert space.

It is very easy to see that a Cauchy sequence in $H$ converges element-wise to an element $x^*$. But how would one show that this element is in $H$, i.e. that $\|x\|^2 < \infty$?

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Can you do the special case $H_{n} = \mathbb{R}$ for all $n$? What you get is the usual space $\ell^2$ and the argument for the general case is exactly the same. –  t.b. Oct 4 '11 at 11:13
And your final result is called the "orthogonal direct sum" of the Hilbert spaces. –  GEdgar Oct 4 '11 at 12:06
Of course another part of the proof that $H$ is a Hilbert space is writing down an inner product corresponding to this norm. –  GEdgar Oct 4 '11 at 12:07
or verifying the parallelogram identity. –  Mark Oct 4 '11 at 22:34

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