Inner product

For a vector space $\displaystyle V$ over $F \subseteq \mathbb{C}$ (or $\mathbb{R}$ ), an inner product is a binary operation $\langle \cdot, \cdot \rangle : V \times V \mapsto \mathbb{C}$ (or $\mathbb{R}$ ) which satisfies the following axioms:

For all $\mathbf{v, w} \in V$ , $\langle \mathbf{ v,w } \rangle = \overline{ \langle \mathbf{ w,v } \rangle }$ .
For all $\alpha \in F$ , $\mathbf{v,w} \in V$ , $\langle \alpha \mathbf{v,w} \rangle = \alpha\langle \mathbf{v,w} \rangle$ .
For all $\mathbf{u,v,w} \in V$ , $\langle \mathbf{ u+v, w} \rangle = \langle \mathbf{u,w} \rangle + \langle \mathbf{ v, w } \rangle$ .

From these three axioms we can also conclude that $\langle \mathbf{v}, \alpha\mathbf{w} \rangle = \alpha \langle \mathbf{v,w} \rangle$ and $\langle \mathbf{ v, u+w} \rangle = \langle \mathbf{v,u} \rangle + \langle \mathbf{v,w} \rangle$ .

For all $\mathbf{v} \in V$ , $\langle \mathbf{v, v} \rangle \ge 0$ , with equality if and only if $\mathbf{v} = \mathbf{0}$ .

This is reasonable because from the first axiom, we must have $\langle \mathbf{v,v} \rangle \in \mathbb{R}$ .

Note that from these axioms we may also obtain the following result:

$\langle \mathbf{v,w} \rangle = 0$ for all $\mathbf{w}\in V$ if and only if $\mathbf{v = 0}$ .

This is occasionally listed as an axiom in place of the condition that equality holds on the condition $\langle \mathbf{v,v} \rangle = 0$ exactly when $\mathbf{v=0}$ .

Examples

For the vector space $\mathbb{R}^n$ , the dot product is perhaps the most familiar example of an inner product.

In addition, for the vector space $\displaystyle A$ of functions mapping some interval $I \mapsto \mathbb{R}$ , the operator $\displaystyle \int_{I} f(x)g(x) dx$ is an inner product for $f, g \in A$ .

Resources

Cauchy-Schwarz Inequality

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Inner product

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