My book in functional analysis states that "the dual of $L^p$ is $L^q$". Is this just a short way of saying that $(L^p)^{'}$ is isometrically isomorphic to $L^q$? I mean, if $\ell \in (L^p)^{'}$ then $\ell$ is of the form $\ell=\int u(t)dt$, $u(t) \in L^q$, while $L^q$ consists of functions. How could these spaces be equal?
2026-03-26 18:50:42.1774551042
How can it be true that the dual of Lp is Lq...?
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Saying "$(L^p)'$ is $L^q$" means they are (isometrically) isomorphic and it is clear which isomorphism should be used. Strictly speaking that is sloppy language, so you are right. But it is very common practice in many branches of mathematics to just use "is" or "equal" when there is a canonical isomorphism between two objects, even if they are formally different.