Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $E \subset H$ be an orthonormal set, and let $x \in H$. The set \begin{align*} \{e \in E : (x,e)_H \ne 0\} \end{align*} is countable.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space) and let $E \subset H$ be an orthonormal set. The following conditions are equivalent. 1. $E$ is complete. 2. $\overline{\operatorname{span}}(E)=H$. 3. For every $x \in H$, \begin{align*} \|x\|_H^2 = \sup_{F \subset E,\ F \text{ finite}} \sum_{e \in F}|(x,e)_H|^2. \end{align*} 4. For every $x \in H$, the net of finite partial sums \begin{align*} \sum_{e \in F}(x,e)_H e, \end{align*} indexed by finite subsets $F \subset E$, converges to $x$ in $H$.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space). If $E$ and $F$ are orthonormal bases of $H$, then $E$ and $F$ have the same cardinality.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space) with [orthonormal basis](/page/Orthonormal%20Basis) $E$, and let $(\lambda_e)_{e\in E}$ be a bounded family of scalars. The formula \begin{align*} Tx = \sum_{e\in E}\lambda_e (x,e)_H e \end{align*} defines a [bounded linear operator](/page/Bounded%20Linear%20Operator) $T\in\mathcal L(H)$, and \begin{align*} \|T\|_{\mathcal L(H)} = \sup_{e\in E}|\lambda_e|. \end{align*}

Let $X$ be a set and let $\mathcal A$ be an algebra of sets on $X$. The smallest monotone class containing $\mathcal A$ is $\sigma(\mathcal A)$.

Let $(X,\mathcal F)$ and $(Y,\mathcal G)$ be measurable spaces, and let $\mathcal F\otimes\mathcal G$ be the product sigma-algebra on $X\times Y$. The projections \begin{align*} \pi_X:X\times Y &\to X,\\ \pi_Y:X\times Y &\to Y \end{align*} are measurable.

Let $(X,d)$ be a separable [metric space](/page/Metric%20Space). Then $\mathcal B(X)$ is countably generated.

Let $X$ be a set and let $(\mathcal F_i)_{i\in I}$ be a nonempty family of sigma-algebras on $X$. Then \begin{align*} \bigcap_{i\in I}\mathcal F_i \end{align*} is a sigma-algebra on $X$.

Let $(X,d_X)$ and $(Y,d_Y)$ be separable metric spaces, and equip $X\times Y$ with the [product topology](/page/Product%20Topology). Then \begin{align*} \mathcal B(X\times Y)=\mathcal B(X)\otimes\mathcal B(Y). \end{align*}

Let $(X,d)$ be a [metric space](/page/Metric%20Space). The Borel sigma-algebra $\mathcal B(X)$ is generated by the open balls $B(x,r)$ with $x \in X$ and $r>0$. If $X$ is separable, then $\mathcal B(X)$ is generated by the balls $B(x,r)$ with $x$ in a fixed countable [dense subset](/page/Dense%20Subset) of $X$ and $r \in \mathbb Q$ with $r>0$.