Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now consider $N_T(W)$, the normalizer in $T$ of $W$. I would like to be sure that this is a closed connected subgroup of $T$; i.e. I'd like it to be a torus. Is this true in general?
Here's a more specific question (supposing that the first question is too hard, or is not true in general). Consider $G=UT$ a solvable linear algebraic subgroup of $GL_r$. Suppose that $U_1$ is some subgroup of $U$. I'd like to be sure that $N_T(U_1)$ is a subtorus of $T$. I'm most interested in the situation when $p$ is large (I can imagine that this statement may only be true for $p>r$, say, but this is fine for the application I have in mind.)