If $V$ is a finite-dimensional inner product space and $T: V\to V$ is an invertible, positive-definite (i.e., $\forall v \in V\backslash \{0\}, \langle Tv, v\rangle > 0$) linear map, prove that $T^2 = T^*T$ where $T^*$ is the adjoint of $T$.
In the case where $V$ is over the field $\mathbb{C}$, T is self-adjoint, but my proof of this relies on the fact: \begin{equation} \left(\forall v \in V, \langle Av, v\rangle = 0\right) \implies A=0 \end{equation} which I've read doesn't hold in the real case.
Is there an alternative approach if $V$ is over the field $\mathbb{R}$, or better yet an approach that works over both $\mathbb{R}$ and $\mathbb{C}$?