Complex Inner Products

Why

What is an inner product if we take a vector space over the complex numbers.

Definition

An inner produce over a complex vector space is positive definite, Hermitian, and linear in the first argument.

Notation

Let $(V, \C )$ be a complex vector space. Let $f: V \times V \to C$ be a function such that

1. $f(x, x) \geq 0$, $f(x, x) = 0 \Leftrightarrow x = 0$;
2. $f(x, y) = \overline{f(y, x)}$
3. $f(ax + by, z) = a(x, z) + b(y, z)$