Is $(V_1\otimes\cdots\otimes V_k)^\ast \simeq V_1^\ast\otimes \cdots \otimes V_k^\ast$ true for infinite dimensional spaces?

The situation here is a little subtle.

The canonical inclusion $(V^*_1\otimes\dots\otimes V_k^*)\hookrightarrow(V_1\otimes\dots\otimes V_k)^*$ isn't an isomorphism (it's a strict injection), but (assuming the axiom of choice) the spaces are nevertheless isomorphic by some other (non-canonical) map.

The proof that the canonical map isn't an isomorphism is explained in detail in this answer to another question.

The proof that they are isomorphic works by just calculating the dimensions on both sides and finding that they are the same. The dimensions are infinite cardinals. That proof is given here.