For questions involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.
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
- $\langle x,x\rangle =0\Longrightarrow x=0$ and $\langle x,x\rangle\geqslant 0$ for all $x\in H$,
- $(\forall x,y\in H):\langle x,y\rangle=\overline{\langle y,x\rangle}$,
- $(\forall x_1,x_2,y\in H)(\forall\alpha_1,\alpha_2\in\mathbb C):\langle \alpha_1 x_1+\alpha_2 x_2,y\rangle=\alpha_1\langle x_1,y\rangle+\alpha_2\langle x_2,y\rangle$.
The map $\lVert\cdot\rVert\colon H\to\mathbb R_+$, defined by $\lVert x\rVert =\langle x,x\rangle^{\frac 12}$ is a norm.
If $(H,\lVert \cdot\rVert)$ is complete, then $H$ is called a Hilbert space.
Example: The space $H$ of all sequences $x_0,x_1,x_2,\ldots$ of complex numbers such that $\sum_{n=0}^\infty|x_n|^2<\infty$, with the inner product $$\bigl\langle(x_0,x_1,x_2,\ldots),(y_0,y_1,y_2,\ldots)\bigr\rangle =\sum_{n=0}^{+\infty}x_n\overline{y_n}$$is a Hilbert space.
