Let $G$ be the points of a connected, reductive group over $\mathbb R$, and let $K$ be a maximal compact subgroup of $G$. Let $(\pi,V)$ be a continuous representation of $G$ on a Hilbert space for which $\pi|_K$ is unitary. For each continuous irreducible representation $\tau$ of $K$ (automatically finite dimensional), the $\tau$-isotypic subspace $S_{\tau}$ is defined to be the linear span of all $K$-subspaces of $V$ which are isomorphic to $\tau$. The representation $\pi$ is called admissible if each $S_{\tau}$ is finite dimensional.
Then $V$ is a Hilbert space direct sum of representations of $K$:
$$V = \hat{\bigoplus\limits_{\tau \in \hat{K}}} S_{\tau}, v = \sum\limits_{\tau \in \hat{K}} v_{\tau}$$
An element $v$ of $V$ is called $K$-finite if the $K$-subspace generated by $v$ is finite dimensional.
Question: Obviously if $v_{\tau} = 0$ for almost all $\tau$, then $v$ is $K$-finite. Is the converse true?
It seems like it should be true, but I can't seem to be able to prove this.
For my second question, let $V_0$ be the space of $K$-finite vectors in $V$. It is norm dense and stable under the action of the complexified Lie algebra $\mathfrak g_{\mathbb C} = \operatorname{Lie}(G) \otimes_{\mathbb R} \mathbb C$. I've heard that there is a bijection between $\mathfrak g_{\mathbb C}$-$K$ invariant subspaces $U_0$ of $V_0$ and closed $G$-invariant subspaces $U$ of $V$, given by
$$U \mapsto U \cap V_0$$
$$U_0 \mapsto \overline{U_0}$$
One composition, $U = \overline{(U \cap V_0)}$ is obvious. The other way, $U_0 = \overline{U_0} \cap V_0$, is not clear to me. To prove the inclusion '$\supseteq$', it seems we need to prove:
If $U_0 \subset V_0$ is dense in $V$, $\mathfrak g_{\mathbb C}$-stable, and $K$-stable, then $U_0 = V_0$.
The positive answer to the first question follows formally:
Let $W\subseteq V$ be a finite-dimensional $K$-invariant subspace. By representation theory of compact groups, $W$ decomposes into direct sum of irreducible subrepresentations, and listing these up to iso as $\tau_1, \tau_2, \dots \tau_n$, we have $W=W_{\tau_1}\oplus W_{\tau_2} \oplus \dots \oplus W_{\tau_n},$ where $W_{\tau_i}$ is the $\tau_i$-isotypical component of $W$. But then $W_{\tau_i} \subseteq S_{\tau_i}, $ and so $W \subseteq S_{\tau_1}\oplus S_{\tau_2} \oplus \dots \oplus S_{\tau_n}$. Now if $W$ is taken as the $K$-subspace generated by a $K$-finite vector $v$, the claim follows.