This is an exercise of an old test of functional analysis
Let $p\in[1,\infty)$ and $(y_n)$ a numerical sequence such that $\displaystyle\sum_{n=1}^\infty x_n y_n<\infty$ for all $(x_n)\in\ell^p$. Show that $(y_n)\in\ell^q$ with $\frac{1}{p}+\frac{1}{q}=1$. Hint: Put $\varphi_j(x)=\displaystyle\sum_{n=1}^j x_n y_n$ .
From the hint I can get the $\varphi_j$ are in $(\ell^p)^*=\ell^q$ and converges to $\varphi(x)=\displaystyle\sum_{n=1}^\infty x_n y_n<\infty$ and $\varphi$ also lies on $\ell^q$ but i can't say why $(y_n)\in\ell^q$.
Can anyone help me?
