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.

[edit] Examples of inner product spaces

$\langle x,y \rangle :=\sum_{k=1}^{n}x_k y_k,\quad\forall x=(x_1,\ldots,x_n)\;\hbox{and}\;y=(y_1,\ldots,y_n) \in \mathbb{R}^n$ .

This inner product induces the Euclidean norm ||x|| = ⟨ x, x ⟩^½.

The space $L^2(\mathbb{R})$ of the equivalence classes of all complex-valued Lebesgue measurable scalar square integrable functions on ℝ with the complex inner product