In mathematics, an inner product is an abstract notion on general vector spaces that is a generalization of the concept of the dot product in the Euclidean spaces. Among other things, the inner product on a vector space makes it possible to define the geometric operation of projection onto a closed subspace (in the metric topology induced by the inner product), just like how the dot product makes it possible to define, in the Euclidean spaces, the projection of a vector onto the subspace spanned by a set of other vectors. The projection operation is a powerful geometric tool that makes the inner product a desirable convenience, especially for the purposes of optimization and approximation.
Formal definition of inner product
Let X be a vector space over a sub-field F of the complex numbers. An inner product on X is a sesquilinear map from to with the following properties:
- and (linearity in the first slot)
- and (anti-linearity in the second slot)
- (in particular it means that is always real)
Properties 1 and 2 imply that .
Note that some authors, especially those working in quantum mechanics, may define an inner product to be anti-linear in the first slot and linear in the second slot, this is just a matter of preference. Moreover, if F is a subfield of the real numbers then the inner product becomes a bilinear map from to , that is, it becomes linear in both slots. In this case the inner product is said to be a real inner product (otherwise in general it is a complex inner product).
Norm and topology induced by an inner product
The inner product induces a norm on X defined by . Therefore it also induces a metric topology on X via the metric .
- ↑ T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York (1982), ISBN 0-387-90666-5 p. 49