Let $X$, $Y$, and $Z$ be normed vector spaces. If $T: X \to Y$ is compact, $A: Z \to X$ is bounded and linear, and $B: Y \to Z$ is bounded and linear, then $T A: Z \to Y$ and $B T: X \to Z$ are compact.

Let $X$ be a [normed vector space](/page/Normed%20Vector%20Space) and let $Y$ be a [Banach space](/page/Banach%20Space). Suppose $T_m: X \to Y$ is compact for every $m \in \mathbb{N}$ and \begin{align*} \|T_m - T\|_{\mathcal{L}(X,Y)} &\to 0 \end{align*} for a [bounded linear operator](/page/Bounded%20Linear%20Operator) $T: X \to Y$. Then $T$ is compact.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space). An orthonormal system in $H$ is maximal if and only if it is an [orthonormal basis](/page/Orthonormal%20Basis) of $H$.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space) over $\mathbb{R}$ or $\mathbb{C}$, and let $(e_i)_{i \in I}$ be an [orthonormal basis](/page/Orthonormal%20Basis) of $H$, meaning that $(e_i)_{i \in I}$ is an orthonormal family and \begin{align*} \overline{\operatorname{span}\{e_i : i \in I\}} = H. \end{align*} Define $\ell^2(I)$ to be the Hilbert space of scalar families $a = (a_i)_{i \in I}$ such that \begin{align*} \|a\|_{\ell^2(I)}^2 := \sup\left\{\sum_{i \in F} |a_i|^2 : F \subset I \text{ finite}\right\} < \infty. \end{align*} Then the coordinate map \begin{align*} U: H &\to \ell^2(I) \\ x &\mapsto \bigl((x,e_i)_H\bigr)_{i \in I} \end{align*} is a linear isometric isomorphism. Its inverse is the map \begin{align*} V: \ell^2(I) &\to H \\ (a_i)_{i \in I} &\mapsto \lim_{F} \sum_{i \in F} a_i e_i, \end{align*}…

The sine system $(e_k)_{k\ge 1}$ is an [orthonormal basis](/page/Orthonormal%20Basis) of $L^2((0,\pi))$.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $e_1,\ldots,e_n$ be orthonormal vectors in $H$, and let $a_1,\ldots,a_n$ be scalars. Then \begin{align*} \left\|\sum_{k=1}^n a_k e_k\right\|_H^2=\sum_{k=1}^n|a_k|^2. \end{align*}

Let $(\Omega,\mathcal F)$ be a measurable space, and let $(E_i,\mathcal E_i)$ be measurable spaces for $i \in \{1,\dots,n\}$. A map \begin{align*} X: \Omega &\to E_1 \times \cdots \times E_n \end{align*} with coordinate maps $X_i: \Omega \to E_i$ is measurable from $(\Omega,\mathcal F)$ to $(E_1 \times \cdots \times E_n,\mathcal E_1 \otimes \cdots \otimes \mathcal E_n)$ iff each coordinate map $X_i$ is measurable.