Definition:Inner Product

Given a vector space $$V$$ over a field $$\mathbb{F}$$, an inner product is a mapping $$\langle \cdot, \cdot \rangle : V \times V \rightarrow \mathbb{F}$$ that satisfies the following properties:
 * 1) $$\langle x, y \rangle = \overline{ \langle y, x \rangle}$$, commonly referred to as conjugate symmetry.
 * 2) $$\langle a x, y \rangle = a \langle x, y \rangle$$
 * 3) $$\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle $$
 * 4) $$\langle x, x \rangle \geq 0$$ and $$\langle x, x\rangle = 0$$ iff $$x = 0$$

Inner Product Space
An inner product space is a vector space together with its associated inner product. It is also sometimes known as a pre-Hilbert space because with the addition of the metric induced by the inner product, it becomes a Hilbert space.

Dot Product
The most well known example of an inner product is the dot product (see proof here).