Revision as of 13:38, 4 January 2009 by imported>Richard Pinch
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
endowed with the real inner product
for all
. This inner product induces the Euclidean norm ![{\displaystyle \|x\|=\langle x,x\rangle ^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0ea2ce90aea85360c49f1aec885e121f7e5dc6)
- 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 ![{\displaystyle \|f\|=\left(\int _{-\infty }^{\infty }|f(x)|^{2}dx\right)^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de3abb0a97d8c87467e5cadadc083b10ff59a408)