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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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