Let $H$ a vector space over the field $\mathbb C$, and $\langle \cdot,\cdot\rangle\colon H\times H\to \mathbb{C}$ a map which satisfies

  1. $\langle x,x\rangle =0\Rightarrow x=0$ and $\langle x,x\rangle\geq 0$ for all $x\in H$,
  2. $\forall\, x,y\in H\quad \langle x,y\rangle=\overline{\langle y,x\rangle}$,
  3. $\forall\, x_1,x_2,y\in H,\alpha_1,\alpha_2\in\mathbb C\quad \langle \alpha_1 x_1+\alpha_2 x_2,y\rangle=\alpha_1\langle x_1,y\rangle+\alpha_2\langle x_2,y\rangle$.

The map $\lVert\cdot\rVert\colon H\to\mathbb R_+$, defined by $\lVert x\rVert =\left(\langle x,x\rangle\right)^{\frac 12}$ is a norm.

If $(H,\lVert \cdot\rVert)$ is complete, then $H$ is called a Hilbert space.

Example: The space $H:=\lbrace x\in \mathbb{C}^n\mid \sum_{n=1}^{+\infty}|x_n|^2<\infty\rbrace$ with the inner product $\langle x,y\rangle =\sum_{n=1}^{+\infty}x_n\overline{y_n}$ is a Hilbert space.

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