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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

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 $\mathbb {R}$ then the inner product becomes a bilinear map from $X\times X$ to $\mathbb {R}$ , 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

The inner product induces a norm $\|\cdot \|$ on X defined by $\|x\|=\langle x,x\rangle ^{1/2}$ . 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

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

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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]].
+{{subpages}}
+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]] 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 [[spanning set|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 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 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>
+#<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>
 Properties 1 and 2 imply that <math>\langle x,y\rangle=0\, \forall x \in X \Rightarrow y=0</math>.
-Note that some authors 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 <math>\mathbb{R}</math> then the inner product becomes a ''bilinear'' map from <math>X \times X </math> to <math>\mathbb{R}</math>, that is, it becomes linear in both slots.
+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 <math>\mathbb{R}</math> then the inner product becomes a ''bilinear'' map from <math>X \times X </math> to <math>\mathbb{R}</math>, 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]]  <math>\|\cdot\|</math> on ''X'' defined by <math>\|x\|=\langle x,x \rangle^{1/2}</math>. Therefore it also induces a [[metric space#metric topology|metric topology]] on ''X'' via the metric <math>d(x,y)=\|x-y\|</math>.
+The inner product induces a [[norm (mathematics)|norm]]  <math>\|\cdot\|</math> on ''X'' defined by <math>\|x\|=\langle x,x \rangle^{1/2}</math>. Therefore it also induces a [[metric space#metric topology|metric topology]] on ''X'' via the metric <math>d(x,y)=\|x-y\|</math>.
+==Reference ==
-==See also==
+{{reflist}}[[Category:Suggestion Bot Tag]]
-[[Inner product space]]
-[[Hilbert space]]
-[[Norm]]
-[[Category:Mathematics_Workgroup]]
-[[Category:CZ Live]]

Inner product: Difference between revisions

Latest revision as of 11:00, 1 September 2024

Formal definition of inner product

Norm and topology induced by an inner product

Reference

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