Rayleigh Quotient of a Self-Adjoint Operator

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $T$ be a linear operator on $V$ that is self-adjoint, and let $S$ be the set of unit vectors of $V$.

The Rayleigh quotient of $T$ is the map $R_{T}:S\to\mathbb{R}$ sending each $x\in S$ to

$$R_{T}(x)=\langle x,T(x)\rangle ,$$

which is a real number by claim 1 of Elementary Properties of a Self-Adjoint Operator.