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.

## Examples of inner product spaces[edit]

1. The Euclidean space R n {\displaystyle \mathbb {R} ^{n}} endowed with the real inner product  ⟨ x , y ⟩ = ∑ k = 1 n x k y k {\displaystyle \langle x,y\rangle =\sum _{k=1}^{n}x_{k}y_{k}} for all  x = ( x 1 , … , x n ) , y = ( y 1 , … , y n ) ∈ R n {\displaystyle x=(x_{1},\ldots ,x_{n}),y=(y_{1},\ldots ,y_{n})\in \mathbb {R} ^{n}} . This inner product induces the Euclidean norm  ‖ x ‖ = ⟨ x , x ⟩ 1 / 2 {\displaystyle \|x\|=\langle x,x\rangle ^{1/2}}
2. The space  L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} of the equivalence classes of all complex-valued Lebesgue measurable scalar square integrable functions on  R {\displaystyle \mathbb {R} } with the complex inner product  ⟨ f , g ⟩ = ∫ − ∞ ∞ f ( x ) g ( x ) ¯ d x {\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}dx} . Here a square integrable function is any function f satisfying  ∫ − ∞ ∞ | f ( x ) | 2 d x < ∞ {\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}dx<\infty } . The inner product induces the norm  ‖ f ‖ = ( ∫ − ∞ ∞ | f ( x ) | 2 d x ) 1 / 2 {\displaystyle \|f\|=\left(\int _{-\infty }^{\infty }|f(x)|^{2}dx\right)^{1/2}}