Definition:Inner Product

Definition

Given a vector space $V$ over a subfield $\mathbb F$ of $\C$, an inner product is a mapping $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ that satisfies the following properties ($\forall x,y,z \in V, a \in \Bbb F$):

$(1): \qquad \left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle}$, commonly referred to as conjugate symmetry

$(2): \qquad \left \langle {a x, y} \right \rangle = a \left \langle {x, y} \right \rangle$

$(3): \qquad \left \langle {x + y, z} \right \rangle = \left \langle {x, z} \right \rangle + \left \langle {y, z} \right \rangle $

$(4): \qquad \left \langle {x, x} \right \rangle \ge 0$

$(5): \qquad \left \langle {x, x} \right \rangle = 0$ iff $x = \mathbf{0}_V$

That is, an inner product is a semi-inner product together with the extra condition $(5)$.

If $\mathbb F$ is a field not contained in $\C$ then $(1)$ above is taken to be:

$(1^\prime): \qquad \left \langle {x, y} \right \rangle = \left \langle {y, x} \right \rangle$, in words, an inner product is symmetric.

Inner Product Space

An inner product space is a vector space together with an 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).

See also

• Semi-inner Product, a slightly more general concept.