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

1. The Euclidean space $\mathbb {R} ^{n}$ endowed with the real inner product $\langle x,y\rangle =\sum _{k=1}^{n}x_{k}y_{k}$ for all $x=(x_{1},\ldots ,x_{n}),y=(y_{1},\ldots ,y_{n})\in \mathbb {R} ^{n}$. This inner product induces the Euclidean norm $\|x\|=\langle x,x\rangle ^{1/2}$
2. The space $L^{2}(\mathbb {R} )$ of the equivalence classes of all complex-valued Lebesgue measurable scalar square integrable functions on $\mathbb {R}$ 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 $\int _{-\infty }^{\infty }|f(x)|^{2}dx<\infty$. The inner product induces the norm $\|f\|=\left(\int _{-\infty }^{\infty }|f(x)|^{2}dx\right)^{1/2}$