Inner product space: Difference between revisions

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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.

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 $\|x\|=<x,x>^{1/2}$
The space $L^{2}(\mathbb {R} )$ of the equivalence class 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 $\|f\|=\left(\int _{-\infty }^{\infty }|f(x)|^{2}dx\right)^{1/2}$

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 ==Examples of inner product spaces==
 #The Euclidean space <math>\mathbb{R}^n</math> endowed with the real inner product <math>\langle x,y \rangle =\sum_{k=1}^{n}x_k y_k</math> for all <math>x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n) \in \mathbb{R}^n</math>. This inner product induces the Euclidean  norm <math>\|x\|=<x,x>^{1/2}</math>
-#The space <math>L^2(\mathbb{R})</math> of the [[equivalence class]] of all complex-valued [[Lebesgue measure|Lebesgue]] [[measurable function|measurable]] scalar square [[integrable function]]s on <math>\mathbb{R}</math> with the complex inner product <math>\langle f,g\rangle =\int_{-\infty}^{\infty} f(x)\overline{g(x)}dx</math>. Here a square integrable function is any function ''f'' satisfying <math>\int_{-\infty}^{\infty} |f(x)|^2dx<\infty</math>. The inner product induces the norm <math>\|f\|=\left(\int_{-\infty}^{\infty} |f(x)|^2dx\right)^{1/2}</math>
+#The space <math>L^2(\mathbb{R})</math> of the [[equivalence class]] of all complex-valued [[Lebesgue measure|Lebesgue]] [[measurable function|measurable]] scalar square integrable functions on <math>\mathbb{R}</math> with the complex inner product <math>\langle f,g\rangle =\int_{-\infty}^{\infty} f(x)\overline{g(x)}dx</math>. Here a square integrable function is any function ''f'' satisfying <math>\int_{-\infty}^{\infty} |f(x)|^2dx<\infty</math>. The inner product induces the norm <math>\|f\|=\left(\int_{-\infty}^{\infty} |f(x)|^2dx\right)^{1/2}</math>