If $(a_n)$ is such that $\sum_{n=1}^\infty a_nb_n$ converges for every $b\in\ell_2$, then $a\in\ell_2$

Question

Please help me with this question. I've been thinking about it for almost two days. Let $a_n$ a real series that have the following property: for every series $b_n$ in $l_2$: $\sum_{n=1}^\infty a_nb_n$ converges. prove that $a_n$ in $l_2$.

Answer 1

I figured this out thanks to a tip from my professor to use the uniform boundedness principle.

(1) Define $T_N : \ell^2(\mathbb{C}) \rightarrow \mathbb{C}$ by $$ T_N(y) = \sum\limits_{i=1}^N x_i y_i $$ which are each bounded linear functionals by Cauchy-Schwarz.

(2) Apply the uniform boundedness principle to the family $\{T_N\}_{N \in \mathbb{N}}$. The hypothesis ensures that this family is uniformly bounded.

(3) The linear map $T$ defined by $T(y)= \lim\limits_{N\rightarrow \infty} T_N(y)$ is bounded as well by (2), specifically, $$ \|T\| := \sup\limits_{\|y\| \leq 1} \|Ty\| = \sup\limits_{\|y\| \leq 1} \lim\limits_{N\rightarrow\infty} \|T_N(y)\| \leq \sup\limits_{\|y\| \leq 1} M\|y\| = M $$ for $M$ the uniform bound.

(4) The Riesz representation theorem implies that $T$ comes from a sequence in $\ell^2$. This sequence must be $x$.

Answer 2

Using Holder inequality we say that $$\sum_{n} a_n b_n\leq \left(\sum_n |a_n|^p\right)^{1/p}\left(\sum_n |b_n|^2\right)^{1/2} $$ with $1/p+1/2= 1$ and from that we get $p=2.$ So the result.