$L^p$-space isomorphisms Let $G$ be a compact hausdorff abelian group.
There is a Haar integral $\int : C(G) \rightarrow \mathbb{R}$ on $C(G)$, the $C^*$-algebra of continuous functions from $G$ to $\mathbb{R}$. We can complete this $C^*$-algebra in various ways to get $L^p(G)$, the completion of $C(G)$ under the norm $||f||_p = \sqrt[p]{\int_G |f|^p}$.
We have a map
$$C(G) \rightarrow C(\hat{G})$$
sending $f : G \rightarrow \mathbb{R}$ to the map $\hat{f} : G \rightarrow \mathbb{R}$ sending $\chi$ to $\int_G f \chi$.
My question is about completing on one side or another by various norms. For which $p$ and $q$ do we have a map
$$L^p(G) \rightarrow L^q(\hat{G})$$
and for which $p$ and $q$ is this an isomorphism?
I suspect that for $p$ and $q$ such that $1/p + 1/q = 1$, we have such a map, and that it is an isomorphism. But a reference would be nice.
Hint: If $1 &lt; p &lt; \infty$, $L^p$ is reflexive. Otherwise not.
Hint: If $1 &lt; p &lt; \infty$, $L^p$ is reflexive. Otherwise not.