Let $0\neq x=(x_1,x_2,\cdots,x_n,\cdots) \in\ell^4$. For which of the following values of $p$, the series $$\sum_{i=1}^\infty x_iy_i$$ converges for every $y=(y_1,y_2,\cdots,y_n,\cdots) \in\ell^p $ ?
(a) $p=1$ $\hspace{2cm}$ (b) $p=2$$\hspace{2cm}$ (c) $p=3$ $\hspace{2cm}$ (d) $p=4$
I know this result: For any $a=(a_i) \in \ell^q$, the series $\sum_{i=1}^\infty a_iy_i$ converges absolutely, for all $y=(y_i) \in \ell^p$ where $q \in (1,\infty]$ and $p \in [1,\infty)$ with $\frac{1}{p}+\frac{1}{q}=1$
So by above result, $\sum_{i=1}^\infty x_iy_i$ converges absolutely, for all $y=(y_i) \in \large\ell^\frac{4}{3}$. But how about these four options ?
Here $$ \sum_{i=1}^\infty \vert x_iy_i \vert \leq \vert\vert x \vert \vert_{\large\ell^4}\vert\vert y \vert \vert_{\large\ell^p} $$ How to choose suitable $p$ to make the RHS to be small? Any Guidance please?
The correct answer is 1). Use the fact that $p \leq \frac 4 3 $ and $(y_n) \in \ell^{p}$ implies $(y_n) \in \ell^{4/3}$