Inner product space: Difference between revisions

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In [[mathematics]], an '''inner produce space''' is a [[vector space]] that is endowed with an [[inner product]]. It is also a [[normed space]] since an inner product induces a norm on the vector space on which it is defined. A [[completeness|complete]] inner product space is called a [[Hilbert space]].
In [[mathematics]], an '''inner produce space''' is a [[vector space]] that is endowed with an [[inner product]]. It is also a [[normed space]] since an inner product induces a norm on the vector space on which it is defined. A [[completeness|complete]] inner product space is called a [[Hilbert space]].


==Examples of normed spaces==
==Examples of inner product spaces==
#The Euclidean space <math>\mathbb{R}^n</math> endowed with the real inner product <math>\langle x,y \rangle =\sqrt{\sum_{k=1}^{n}x_k y_k}</math> for all <math>x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n) \in \mathbb{R}^n</math>. This inner product induces the Euclidean  norm <math>\|x\|=<x,x>^{1/2}</math>  
#The Euclidean space <math>\mathbb{R}^n</math> endowed with the real inner product <math>\langle x,y \rangle =\sqrt{\sum_{k=1}^{n}x_k y_k}</math> for all <math>x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n) \in \mathbb{R}^n</math>. This inner product induces the Euclidean  norm <math>\|x\|=<x,x>^{1/2}</math>  
#The space <math>L^2(\mathbb{R})</math> of the [[equivalence class]] of all complex-valued [[Lebesque measurable]] scalar square integrable functions on <math>\mathbb{R}</math> with the complex inner product <math>\langle f,g\rangle =\int_{-\infty}^{\infty} f(x)\overline{g(x)}dx</math>. Here a square integrable function is any function ''f'' satisfying <math>\int_{-\infty}^{\infty} |f(x)|^2dx<\infty</math>. The inner product induces the norm <math>\|f\|=\int_{-\infty}^{\infty} |f(x)|^2dx</math>   
#The space <math>L^2(\mathbb{R})</math> of the [[equivalence class]] of all complex-valued [[Lebesque measurable]] scalar square integrable functions on <math>\mathbb{R}</math> with the complex inner product <math>\langle f,g\rangle =\int_{-\infty}^{\infty} f(x)\overline{g(x)}dx</math>. Here a square integrable function is any function ''f'' satisfying <math>\int_{-\infty}^{\infty} |f(x)|^2dx<\infty</math>. The inner product induces the norm <math>\|f\|=\int_{-\infty}^{\infty} |f(x)|^2dx</math>   

Revision as of 04:40, 5 October 2007

In mathematics, an inner produce space is a vector space that is endowed with an inner product. It is also a normed space since an inner product induces a norm on the vector space on which it is defined. A complete inner product space is called a Hilbert space.

Examples of inner product spaces

  1. The Euclidean space endowed with the real inner product for all . This inner product induces the Euclidean norm
  2. The space of the equivalence class of all complex-valued Lebesque measurable scalar square integrable functions on with the complex inner product . Here a square integrable function is any function f satisfying . The inner product induces the norm

See also

Completeness

Banach space

Hilbert space