Let $M=\{(x_n)_n \in {\displaystyle \ell ^{2}}| \sum_{n=1}^{\infty}n^2|x_n|^2 < \infty\}$. How can I show that the interior of M is empty?
I already showed that M is convex but I'm not sure if that's helping.
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What is $\ell$? – Olivier Roche Apr 01 '20 at 19:47
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I would assume its the space of square summable sequences in R, right? – Fede Poncio Apr 01 '20 at 19:50
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Usually written $l_2$. – copper.hat Apr 01 '20 at 19:56
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Yes, I meant the space of square summable sequences in R. – Lea Apr 01 '20 at 20:17
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1It's better than a convex subset. It is a closed subspace .. can it have non empty interior? – r9m Apr 01 '20 at 20:18
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1@r9m: Nice approach.. – copper.hat Apr 01 '20 at 20:29
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Hint:
Let $y_n = {1 \over n}$, note that $y \in l_2$, but $y \notin M$.
Suppose $x \in M$. Look at $x+ty$ for small $t$.
copper.hat
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