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

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