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I'm trying to prove some properties of sequence spaces. I already know that the space $l^{\infty}$ of all bounded sequences isn't an inner product space, isn't separable but it is complete with $sup$ norm.

The space $c$ of all convergent sequences is also complete, as a closed subspace of $l^{\infty}$, and so is $c_0$ - the space off all sequences with entries equal to zero from some point on.

I also know that both $c$ and $c_0$ are separable.

My question is - are $c$ and $c_0$ inner product spaces. Should I look for an example of sequences which don't satisfy the parallelogram law.

Thank you.

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  • $\begingroup$ yes. you are in right direction. $\endgroup$ – GA316 Oct 30 '13 at 6:21
  • $\begingroup$ maybe it is too big a hint, on $R^2$, take the norm $||\cdot||_n=n\sqrt{|x_1|^n+|x_2|^n}$, for what values of $n$ is this an inner product space? all the examples I have seen, not that many, admittedly has always something to do with this magic number $\endgroup$ – Lost1 Oct 30 '13 at 6:24
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Should I look for an example of sequences which don't satisfy the parallelogram law.

Yes you should. Take the sequences $x=(1,0,0,0,\dots)$ and $y=(0,1,0,0,\dots)$, write down $$\|x+y\|^2+\|x-y\|^2\overset{?}{=}2(\|x\|^2+\|y\|^2)$$ and watch it fail.

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