Inner product

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This editable Main Article is under development and subject to a disclaimer.

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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[edit]

Let X be a vector space over a sub-field F of the complex numbers. An inner product ⟨⋅,⋅⟩ on X is a sesquilinear[1] map from X×X to ℂ with the following properties:

1. ⟨x,y⟩=⟨y,x⟩‾∀x,y∈X
2. ⟨x,y⟩=0∀y∈X⇒x=0
3. ⟨αx1+βx2,y⟩=α⟨x1,y⟩+β⟨x2,y⟩ ∀α,β∈F and ∀x1,x2,y∈X (linearity in the first slot)
4. ⟨x,αy1+βy2⟩=α¯⟨x,y1⟩+β¯⟨x,y2⟩ ∀α,β∈F and ∀x,y1,y2∈X (anti-linearity in the second slot)
5. ⟨x,x⟩≥0∀x∈X (in particular it means that ⟨x,x⟩ is always real)
6. ⟨x,x⟩=0⇒x=0

Properties 1 and 2 imply that ⟨x,y⟩=0∀x∈X⇒y=0.

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 ℝ then the inner product becomes a bilinear map from X×X to ℝ, 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[edit]

The inner product induces a norm ‖⋅‖ on X defined by ‖x‖=⟨x,x⟩1/2. Therefore it also induces a metric topology on X via the metric d(x,y)=‖x−y‖.

## Reference[edit]

1. ↑ T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York (1982), ISBN 0-387-90666-5 p. 49