Inner product: Difference between revisions
imported>Aleksander Stos (→Formal definition of inner product: minor clarification) |
imported>Hendra I. Nurdin (some corrections and improvements) |
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In [[mathematics]], an '''inner product''' is an abstract notion on general [[vector space|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 set|closed]] (in the metric topology induced by the inner product) subspace, 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 theory|approximation]]. | In [[mathematics]], an '''inner product''' is an abstract notion on general [[vector space|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 set|closed]] (in the metric topology induced by the inner product) subspace, 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 (mathematics)|optimization]] and [[approximation theory|approximation]]. | ||
==Formal definition of inner product== | ==Formal definition of inner product== | ||
Let ''X'' be a vector space over a [[field|sub-field]] ''F'' of the [[complex number|complex numbers]]. An inner product <math>\langle \cdot,\cdot \rangle</math> on ''X'' is a ''sesquilinear''<ref>T. Kato, ''A Short Introduction to Perturbation Theory for Linear Operators'', Springer-Verlag, New York (1982), ISBN 0-387-90666-5 p. 49 </ref> map from <math>X \times X</math> to <math>\mathbb{C}</math> with the following properties: | Let ''X'' be a vector space over a [[field|sub-field]] ''F'' of the [[complex number|complex numbers]]. An inner product <math>\langle \cdot,\cdot \rangle</math> on ''X'' is a ''sesquilinear''<ref>T. Kato, ''A Short Introduction to Perturbation Theory for Linear Operators'', Springer-Verlag, New York (1982), ISBN 0-387-90666-5 p. 49 </ref> map from <math>X \times X</math> to <math>\mathbb{C}</math> with the following properties: | ||
#<math>\langle x,y\rangle=\overline{\langle y,x\rangle}</math> | #<math>\langle x,y\rangle=\overline{\langle y,x\rangle}\,\,\forall x,y \in X</math> | ||
#<math>\langle x,y\rangle=0\, \forall y \in X \Rightarrow x=0</math> | #<math>\langle x,y\rangle=0\,\, \forall y \in X \Rightarrow x=0</math> | ||
#<math>\langle \alpha | #<math>\langle \alpha x_1+\beta x_2,y\rangle= \alpha \langle x_1, y\rangle+\beta \langle x_2, y\rangle</math> <math>\forall \alpha,\beta \in F</math> and <math>\forall x_1,x_2,y \in X</math> (linearity in the first slot) | ||
#<math>\langle x,\alpha | #<math>\langle x,\alpha y_1+\beta y_2 \rangle= \bar\alpha \langle x, y_1\rangle +\bar\beta \langle x, y_2\rangle</math> <math>\forall \alpha,\beta \in F</math> and <math>\forall x,y_1,y_2 \in X</math> (anti-linearity in the second slot) | ||
#<math>\langle x,x\rangle \geq 0</math> (in particular it means that <math>\langle x,x\rangle</math> is always real) | #<math>\langle x,x\rangle \geq 0\,\, \forall x \in X</math> (in particular it means that <math>\langle x,x\rangle</math> is always real) | ||
#<math>\langle x,x\rangle=0 \Rightarrow x=0</math> | #<math>\langle x,x\rangle=0 \Rightarrow x=0</math> | ||
Revision as of 11:30, 5 October 2007
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 (in the metric topology induced by the inner product) subspace, 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[1] 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.
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 .
Reference
- ↑ T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York (1982), ISBN 0-387-90666-5 p. 49