For two functions $x,y:\omega\to\mathbb R$ let $xy:\omega\to\mathbb R$, $xy:n\mapsto x(n)y(n)$, be their pointwise product.


It is clear that for any elements $x,y\in\ell\_2$ their pointwise product $xy$ is an element of $\ell\_2$.



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> **Problem.** Let $\sum\_i x\_i$ be an unconditionally convergent series in $\ell\_2$. Is it true that for any sequence $\{y\_i\}\_{i\in\omega}\subseteq\{y\in\ell\_2:\|y\|\le 1\}$, the series $\sum\_i x\_iy\_i$ converges in $\ell\_2$?
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A perturbation of an unconditionally convergent series in $\ell_2$
Volodymyr Kadets kindly informed me that the answer to this problem is affirmative. His argument easily generalizes to prove the following



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> **Theorem.** For any $p\in[1,\infty)$, any unconditionally convergent series $\sum\_{n\in\omega}x\_n$ in the Banach space $\ell\_p$ and any sequence $\{y\_n\}\_{n\in\omega}\subseteq\{y\in\ell\_2:\|y\|\le 1\}$, the series $\sum\_{n\in\omega}x\_ny\_n$ converges unconditionally in $\ell\_p$.
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*Proof.* Let $q\in(1,\infty]$ be such that $\frac1q+\frac1p=1$. By Bessaga-Pelczynski Theorem (6.4.3 in [this book](https://link.springer.com/book/10.1007/978-3-0348-9196-7)), to show that the series $\sum\_{n\in\omega}x\_ny\_n$ is unconditionally convergent in $\ell\_p$, it suffices to check that it is weakly absolutely convergent, i.e., for every $v\in\ell\_q=\ell\_p^\*$ we have $\sum\_{n\in\omega}|\langle v,x\_ny\_n\rangle|<\infty$.


Holder's inequality ensures that for every $x\in\ell\_p$ the product $vx$ belongs to $\ell\_1$ and the operator $T\_v:\ell\_p\to\ell\_1$, $T\_v:x\mapsto vx$, is continuous. Then the series $\sum\_{n\in\omega}vx\_n$ is unconditionally convergent in $\ell\_1$.


The absolute summability of the identity inclusion $\ell\_1\to\ell\_2$ (which follows from Grothendieck's Theorem 4.3.2 in [this book](https://link.springer.com/book/10.1007/978-3-0348-9196-7)) implies that $\sum\_{n\in\omega}\|vx\_n\|\_{\ell\_2}<\infty$.


Now observe that Cauchy-Bunyakovsky-Schwartz inequality ensures that
$$\sum\_{n\in\omega}|\langle v,x\_ny\_n\rangle|=\sum\_{n\in\omega}|\langle vx\_n,y\_n\rangle|\le\sum\_{n\in\omega}\|vx\_n\|\_{\ell\_2}\cdot\|y\_n\|\_{\ell\_2}\le\sum\_{n\in\omega}\|vx\_n\|\_{\ell\_2}<\infty.$$


