Inner product

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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 continuous 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


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