An infinite completely normal basis theorem

Let $L/K$ 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 $φ:C(G,K)\