What is Hermitian inner product?

A Hermitian inner product on a complex vector space $E$ is a map $\langle\cdot,\cdot\rangle \colon E \times E \to \C$ that is linear in the second variable, conjugate-symmetric ($\langle y, x\rangle = \conj{\langle x, y\rangle}$ — hence conjugate-linear in the first variable), and positive definite ($\langle x, x\rangle > 0$ for $x \neq 0$). The standard example on $\C^n$:

on continuous functions, $\langle f, g\rangle = \int_a^b \conj f\,g$. Norm: $\norm x = \sqrt{\langle x,x\rangle}$; a finite-dimensional complex space so equipped is a Hermitian space.