An infinite completely normal basis theorem

Let $L\mathrm{/}K$ be an arbitrary Galois extension with group $G$, equipped with the Krull topology. Let $C(G,K)$ denote the $K$-vector space of continuous functions $G\to K$, with $K$ discrete and $G$ acting by left translation. We prove that there exists a $G$-equivariant $K$-linear isomorphism $\phi :C(G,K)\to L$ such that, for every open subgroup $H$ of $G$, the map ${\phi }^H:C(H,L^H)\to L$ induced by extension by zero and extension of scalars is an $H$-equivariant $L^H$-linear isomorphism. This gives a completely normal analogue of Lenstra’s infinite normal basis theorem; the case of finite base fields was previously established in an equivalent form by Hachenberger.