Inner product space
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
- The Euclidean space ℝn endowed with the real inner product
- This inner product induces the Euclidean norm ||x|| = 〈 x, x 〉½.
- The space of the equivalence classes of all complex-valued Lebesgue measurable scalar square integrable functions on ℝ with the complex inner product
- Here a square integrable function is any function f satisfying
- The inner product induces the norm
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