Integral Operator bounded on $L^p$, or infinite-dimensional matrix operator bounded on $\ell^p$

Note: As a newer user, forgive me if this is against forum etiquette to bring attention to old (one of which is unanswered) posts. To make this question more novel to the site, I expound on why I did not understand the answer to the probelm involving sequences. If a question like this should not be posted, please mark it as duplicate.

I am having difficulty solving both of these problems (both can be solved similarly, one is just in a function space, the other in a sequence space).

Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Show that the map $A : l^p \rightarrow l^q$ is a bounded linear map

At first both of these problems appear to be a straightforward application of the Closed Graph Theorem. The next obvious step is to use Holder's inequality, but showing that the infinite dimensional matrix or the integral kernel are in the Holder dual space isn't clear (I mean, if we do that anyway, I think we have solved the problem because $(L^p)^* \cong L^q$ and similarly $(\ell^p)^* \cong \ell^q$, with $q$ the Holder conjugate). The second link seems to indicate that the Baire category theorem might be useful to prove the matrix is in $\ell^q$, but I don't know why or where to start. Any hints? Thanks.

• If it helps, these problems are both in Conway's A Course in Functional Analysis, Chapter III, Section 14. – mathishard.butweloveit Jul 19 '18 at 21:21
• My apologies, section 12, not 14. – mathishard.butweloveit Jul 24 '18 at 14:59