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

\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|| = &lang; x, x &rang;&frac12;.


 * 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

\langle f,g\rangle =\int_{-\infty}^{\infty} f(x)\overline{g(x)}dx $$.
 * Here a square integrable function is any function f satisfying

\int_{-\infty}^{\infty} |f(x)|^2dx<\infty $$.
 * The inner product induces the norm $$\|f\|=\left(\int_{-\infty}^{\infty} |f(x)|^2dx\right)^{1/2}$$