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Let $X$ be the vector space of all real sequences with finite support (i.e., there are only finitely many non-zero elements) with the scalar product $$ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i $$ for all $x=(x_1, x_2, \ldots, x_n, \ldots), y=(y_1, y_2, \ldots, y_n, \ldots)\in X$. Checking $X$ is a Hilbert space or not?

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up vote 2 down vote accepted

HINT: Let

$$x^{(n)}_k=\begin{cases} \frac1k,&\text{if }1\le k\le n\\\\ 0,&\text{if }k>n\;. \end{cases}$$

Is $\left\langle x^{(n)}:n\in\Bbb Z^+\right\rangle$ Cauchy? Has it a limit in $X$?

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Dear Sir. Thank you for your interesting example. –  blindman Oct 11 '12 at 9:56
    
@blindman: You’re very welcome. –  Brian M. Scott Oct 11 '12 at 9:57
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