Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 2 down vote accepted


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

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.