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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

  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\|=<x,x>^{1/2}
  2. The space L^2(\mathbb{R}) of the equivalence class of all complex-valued Lebesque 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)|^2dx<\infty. The inner product induces the norm \|f\|=\left(\int_{-\infty}^{\infty} |f(x)|^2dx\right)^{1/2}

See also

Completeness

Banach space

Hilbert space

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