Inner Products

Why

We abstract the notion of inner product to an arbitrary vector space.

Definition

Suppose $\F $ is a field which is either $\R $ or $\C $. Let $(V, \F )$ be a vector space. Then a function $f: V \times V \to \F $ is an inner product on $V$ if

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

A inner product space (or pre-Hilbert space) is a tuple ($V, f)$ where $V$ is an inner product space over $\F $ and $f: V^2 \to \F $ is an inner product.

Notation

Suppose $V$ is a vector space over the field $\F $. We regularly denote an arbitrary inner product for $V$ by $\ip{\cdot ,\cdot }: V^2 \to \F $. So we would denote the inner product of the vector $x$ with the vector $y$ by $\ip{x, y}$.