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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} endowed with the real inner product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x,y \rangle =\sqrt{\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|x\|=<x,x>^{1/2}}
2. The space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2(\mathbb{R})} of the equivalence class of all complex-valued Lebesque measurable scalar square integrable functions on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} with the complex inner product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|f\|=\left(\int_{-\infty}^{\infty} |f(x)|^2dx\right)^{1/2}}

See also

Completeness

Banach space

Hilbert space