Inner product: Difference between revisions

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \cdot,\cdot \rangle} on X is a sesquilinear^[1] map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times X} to $\mathbb {C}$ with the following properties:

$\langle x,y\rangle ={\overline {\langle y,x\rangle }}\,\,\forall x,y\in X$
$\langle x,y\rangle =0\,\,\forall y\in X\Rightarrow x=0$
$\langle \alpha x_{1}+\beta x_{2},y\rangle =\alpha \langle x_{1},y\rangle +\beta \langle x_{2},y\rangle$ $\forall \alpha ,\beta \in F$ and $\forall x_{1},x_{2},y\in X$ (linearity in the first slot)
$\langle x,\alpha y_{1}+\beta y_{2}\rangle ={\bar {\alpha }}\langle x,y_{1}\rangle +{\bar {\beta }}\langle x,y_{2}\rangle$ $\forall \alpha ,\beta \in F$ and $\forall x,y_{1},y_{2}\in X$ (anti-linearity in the second slot)
$\langle x,x\rangle \geq 0\,\,\forall x\in X$ (in particular it means that $\langle x,x\rangle$ is always real)
$\langle x,x\rangle =0\Rightarrow x=0$

Properties 1 and 2 imply that $\langle x,y\rangle =0\,\forall x\in X\Rightarrow y=0$ .

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} then the inner product becomes a bilinear map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times X } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , that is, it becomes linear in both slots.

Norm and topology induced by an inner product

The inner product induces a norm Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\cdot\|} on X defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|x\|=\langle x,x \rangle^{1/2}} . Therefore it also induces a metric topology on X via the metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y)=\|x-y\|} .

Reference

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

@@ Line 1: / Line 1: @@
-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==
 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 x,y\rangle= \alpha \langle x,y\rangle\,\forall \alpha \in F</math> (linearity in the first slot)
+#<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 y\rangle= \bar\alpha \langle x, y\rangle\,\forall \alpha \in F</math> (anti-linearity in the second slot)
+#<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>

Inner product: Difference between revisions

Revision as of 11:30, 5 October 2007

Contents

Formal definition of inner product

Norm and topology induced by an inner product

Reference

See also

Navigation menu

Inner product: Difference between revisions

Revision as of 11:30, 5 October 2007

Formal definition of inner product

Norm and topology induced by an inner product

Reference

See also

Navigation menu

Search