In Silverman's book "The Arithmetic of Elliptic Curves", Lemma 5.8.1 in Chapter II says:
Let $G_K$ act continuously on a $\overline{K}$ vector space $V$ in a way compatible with the normal Galois action on $\overline{K}$. Let $V_K$ be the subspace of vectors fixed by $G_K$, i.e. $$V_K = \{v \in V: v^\sigma = v \text{ for all } \sigma \in G_K\}.$$ Then $V \cong \overline{K} \otimes_K V$, i.e. $V$ has a basis of vectors fixed by $G_K$.
(Here, $K$ is a number field, and $G_K$ its absolute Galois group with the Krull/profinite topology.) I understand almost all of the proof of this lemma, but the following part is giving me a lot of trouble:
The assumption that $G_K$ acts continuously on $V$ means precisely that for any $v \in V$, its stabilizer subgroup $\{\sigma \in G_K:v^\sigma = b\}$ has finite index in $G_K$.
(The book says "finite index in $K$", but I assume that's a typo, and that $G_K$ is what's meant.) The part of the proof before this statement, and the part of the proof after it, I have absolutely no issues with. This statement is really the heart of the proof-- once I understand it, the result reduces to finite Galois theory + a little linear algebra.
First, a clarifying questions. I understand "$G_K$ acts continuously on $V$ in a way compatible with the usual Galois action" to mean "For all $\sigma \in G_K$, $v \mapsto v^\sigma$ is a continuous function from $V$ to $V$, and $(\alpha v)^\sigma = \alpha^\sigma v^\sigma$ whenever $\alpha \in \overline{K}, \sigma \in G_K, v \in V$". Is this correct? If it is, what is the topology on $V$? Silverman doesn't say. Maybe he means the discrete topology, but in that case I must be misunderstanding "continuous", because if $V$ is discrete, asking $v \mapsto v^\sigma$ to be continuous is a meaningless condition (any function from a discrete space is continuous).
Since I either don't know $V$'s topology, or don't know what continuous means, I really don't know how to start. Supposing I did, the natural idea is to reinterpret "finite-index stabilizer" as a statement about the preimage of [whatever thing's continuity we have]. One reason I suspect Silverman means the discrete topology on $V$ is because continuity of an action on a discrete space would have to be checked at each $v$, and the finite-index stabilizer condition is a statement about one individual $v$ at a time.
I think that's just about everything useful I can say about what I've tried so far. I'd appreciate a hint (e.g. the correct definition of continuity here, and maybe some general pointers on how to work with $G_K$'s topology and continuity in this context), and not a full answer, so I can figure this out for myself once I get a proper start.
To say that the action is continuous means that the map $G_K \times V \to V$ is continuous where $G_K$ has the profinite topology and $V$ has the discrete topology, and $G_K \times V$ has the product topology.