# Inner product space

From Knowino

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 ℝ
^{n}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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