Definition:Inner Product

Definition
Given a vector space $V$ over a field $\mathbb F$, an inner product is a mapping $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ that satisfies the following properties:
 * 1) $\left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle}$, commonly referred to as conjugate symmetry.
 * 2) $\left \langle {a x, y} \right \rangle = a \left \langle {x, y} \right \rangle$
 * 3) $\left \langle {x + y, z} \right \rangle = \left \langle {x, z} \right \rangle + \left \langle {y, z} \right \rangle $
 * 4) $\left \langle {x, x} \right \rangle \ge 0$ and $\left \langle {x, x} \right \rangle = 0$ 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 by the Completion Theorem we can extend an inner product space to its completion, so it becomes a Hilbert space.

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