Question about the theorem of highest weights

Question

I have some confusion from reading Theorem 7.3 in Sepanski's Compact Lie groups and would appreciate it if someone could clarify.

1. In part (e) the book says "for $w\in W$, $wV_\lambda=V_{w\lambda}$, so that $\dim V_\lambda=\dim V_{w\lambda}$."

My first question is how do we make sense of the action of the Weyl group $W$ on a weight space $V_{\lambda}$?

(For the proof of part (e), the book refers to the proof of a similar equality for root spaces that I do understand, but I think it is does not carry over from root spaces to weight spaces to give $wV_\lambda=V_{w\lambda}$, no matter how we interpret the left-hand side of this equality.)

2. My second question is about the "connectedness" assumption in the theorem. Isn't this assumption redundant?

Thanks for reading!