Inner product space

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is developing and not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

In mathematics, an inner product 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 classes of all complex-valued Lebesgue 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