Positive Definite Operator

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$, and let $T$ be a linear operator on $V$.

The operator $T$ is positive definite if for every $u\in V$ the complex number $\langle u,T(u)\rangle$ is a real number satisfying

the order being that of the ordered field of real numbers, and if moreover $\langle u,T(u)\rangle=0$ holds only for $u=0_{V}$.