Let $1 \leq p < \infty$ and a sequence $b = (b_j) \in l^p$. Define the operator $T : l^{\infty} \rightarrow l^p$ by $T(x) = (\xi_j b_j )$, where $x = (\xi_j) \in l^{\infty}$. Show that $T$ is well defined, linear, bounded and find $||T||$.
That the operator is linear and bounded this part is done. Well defined operator, I have to show that $\displaystyle\sum_{j=1}^{\infty}|\xi_jb_j|^p<\infty$. This is my attemp:
$\displaystyle\sum_{j=1}^{\infty}|\xi_jb_j|^p\leq \displaystyle\sum_{j=1}^{\infty}|\xi_j|^p\displaystyle\sum_{j=1}^{\infty}|b_j|^p=||b_j||_p^{p}\displaystyle\sum_{j=1}^{\infty}|\xi_j|^p<\infty$.
Is it right?
And the $\lVert T\rVert$, how could I do it? I have to use $||x||=\displaystyle\sup_j|\xi_j|$ or $\lVert x\rVert=\begin{pmatrix}\displaystyle\sum_{j=1}^{\infty}|\xi_j|^p\end{pmatrix}^{1/p}$. Thanks!
The inequality you used is true, but there is no reason for $\sum_j|\xi_j|^p$ to be finite; so the inequality is very coarse. Instead, using that $x\in\ell^\infty$, $$ \|Tx\|^p=\sum_j|\xi_jb_j|^p\leq\sum_j|\xi_j|^p\,|b_j|^p\leq\|x\|_\infty^p\,\sum_j|b_j|^p=\|x\|_\infty^p\,\|b\|_p^p. $$ This shows that $\|T\|\leq\|b\|_p$. Taking $x=1$ the norm is achieved, so $\|T\|=\|b\|_p$.