Inner product space

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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