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A problem asked me to show that $l^{2}$ is normed space with norm

$$\lvert \lvert x \rvert \rvert = \sqrt{\sum^{ \infty }_{ \iota = 1} x^{2}_\iota}.$$

I've shown that this is indeed a norm where the triangle inequality relies on Cauchy-Schwarz.

I am however stuck on the follow up question:

Give an example of two closed linear subspace $A, B$ of $l^{2}$ such that the sum $A + B = \{ a + b: a \in A \quad \land \quad b \in B \}$ is not closed.

Question about latex. While typing in the problem I noticed that typing { and } in math mode does not work using \ { and \ }. How do I type sets on math.stackexchange?

Also I wasn't able to type in $\iota$ as an index on $x_{\iota}$ in the sum above.

Martin Argerami
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simpl0s
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  • I've edited the iota in the subindex with no issue. Not sure what the problem with curly braces is. – Martin Argerami Dec 11 '19 at 03:13
  • @MartinArgerami odd. I looked at your edit and I tried the same and it did not work for me. Perhaps my browser is having some issues. Thanks for the edit. – simpl0s Dec 11 '19 at 03:15
  • \wedge inserts the 'and' operator (if that is what you're asking, I can't quite tell) – Ty Jensen Dec 11 '19 at 03:17
  • @TyJensen When I was typing in the question I was trying to use set notation, i.e. the curly braces to enclose the set. When I was using \ {, the curly braces would just disappear. And the $\iota$ as a subscript on $x_{\iota}$ in the sum would just print the latex code as plain text. Perhaps this issue was only in the preview before posting the question. It seems to not occur after posting. – simpl0s Dec 11 '19 at 03:20
  • Strange, I assume it’s just your browser then. – Ty Jensen Dec 11 '19 at 03:21
  • Voting to close as a duplicate. – Martin Argerami Dec 11 '19 at 03:24

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