Let $x = (x_1, x_2, \ldots )\in l^4$, $x\neq 0$. For which one of the following values of $p$, the series $$ \sum_{i=1}^{\infty} x_i y_i $$ converges for every $y = (y_1, y_2, \ldots )\in l^p$?
- A) 1
- B) 2
- C) 3
- D) 4
What I tried: I tried to use Hölder's inequality, I am almost done, I think I am just missing a little.
$|\sum_{i=1}^{\infty} x_i y_i|\le \sum_{i=1}^{\infty} |x_i y_i|\le ||x||_4 ||y||_p $
where $p$ is the Hölder conjugate of $4$ which gives $p=4/3$.
If $x \in \ell^{q}$ for every $q \in [1,\infty]$ then the series converges in all 4 cases. Assume that $x \in \ell^{4}\setminus \ell^{q}$ for all $q <4$. In this case the answer is A). You already know that the series converges if $y \in \ell ^{4/3}$. If $y \in \ell ^{p}$ with $p \leq \frac 4 3$ then $y \in \ell ^{4/3}$ so the series converges. To show that the series may not converge in cases B), C) and D) take $y_n=sign(x_n)|x_n|^{\alpha}$ for a suitable $\alpha$. I will leave the details to you.