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
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\begin{align*}
\overline{\operatorname{span}\{e_i : i \in I\}} = H.
\end{align*}
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Define $\ell^2(I)$ to be the Hilbert space of scalar families $a = (a_i)_{i \in I}$ such that
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\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*}
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Then the coordinate map
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\begin{align*}
U: H &\to \ell^2(I) \\
x &\mapsto \bigl((x,e_i)_H\bigr)_{i \in I}
\end{align*}
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is a linear isometric isomorphism. Its inverse is the map