Show that if $\sum x_n y_n$ converges for all $y=\{y_n: n\}$ in $\ell_2$ , then $x=\{x_n :n\}$ is in $\ell_2$.

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I am not able to go anywhere with this problem. Please help

First show that the sequence $x_{n}$ is bounded. Now use the Riesz representation theorem. — fiverules, Dec 16 '13 at 04:28

score 2 · Answer 1 · answered Dec 16 '13 at 04:31

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First show that the sequence $x_{n}$ is bounded. If not, then for any $j$ there is $x_{n_{j}}>2^{j}$. Now take $y_{m}=2^{-k}$ if $m=n_{k}$ for some $k$ and $y_{m}=0$ otherwise. Then $\sum x_{n}y_{n} > \infty$, a contradiction. So we get $x_{n}$ is bounded.

Now we got that $L(y)=\sum x_{n}y_{n}$ is a bounded linear functional on $l^{2}$. Use the Riesz representation theorem.

answered Dec 16 '13 at 04:31

fiverules

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I don't know hoe to show that it is a bounded linear functional. – Oliver Dec 16 '13 at 05:59

Show that if $\sum x_n y_n$ converges for all $y=\{y_n: n\}$ in $\ell_2$ , then $x=\{x_n :n\}$ is in $\ell_2$.

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