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A peer of mine recently posed the following question:

When is the functional $f: \ell^p \times \ell^q \to \mathbb R$ given by $f(a_n, b_n) = \sum a_n b_n$ well defined and bounded?

Hölder's Inequality clearly gives this when $p, q$ are conjugates, and there is plenty of theory I can find for specific sequences (for example, if $\sum a_n$ converges and $\sum b_n$ is bounded then $\sum a_n b_n$ converges) but I am unsure if any necessary or sufficient conditions are known if we restrict to sequences from (possibly different) $\ell^p$. Are any such conditions known, or is the resulting space still too large and not tamable?

The problem has an obvious generalization to the $L^p$ spaces, but I wanted to give the question as it was asked.

Brevan Ellefsen
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Assuming that $p, q > 1$ the condition is $p+q \geq pq$. If $\sum a_nb_n$ converges for all $(a_n) \in l^{p}$ then $(b_n) \in l^{p*}$ where $p*$ is the index conjugate to $p$. Thus a necessary condition is $l^{q} \subset l^{p*}$ which implies $q \leq p*$ or $q \leq \frac p {p-1}$. Hence $p+q \geq pq$. Conversely this condition implies convergence of $\sum a_n b _n$ and boundedness of $f$. .

Kavi Rama Murthy
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  • Thank you Kabo. Do you have a reference for the claim "If $\sum a_n b_n$ converges for all $(a_n) \in \ell^p$ then $(b_n) \in \ell^{p^*}$"? I have not come across this result before and it mildly surprises me; feels almost like a converse to Holder. Also, and pardon my ignorance of what might be glaringly obvious, is there anything of interest when either $p,q \in (0,1]$? – Brevan Ellefsen Dec 03 '19 at 08:19
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    @BrevanEllefsen The result I have used is standard and it is found in many books on Functional Analysis. The proof is similar to the following: https://math.stackexchange.com/questions/2364233/prove-if-sum-na-nb-n-converges-whenever-sum-b-n2-lt-infty-then-sum?rq=1 – Kavi Rama Murthy Dec 03 '19 at 08:23