A problem on an operator in inner product space

Question

Let $T \in L(V,W)$ . Define $T^* : W \to V$ by $$\langle T^*w,v\rangle = \langle w,Tv\rangle$$

is it true that $\dim \operatorname{Range}(T^*) = \dim \operatorname{Range}(T)$ ?

Here all the vector spaces are finite dimensional