# Inner product space

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