Orthonormal families, complete orthonormal systems and Hilbert bases

Let $H$ be a real or complex inner-product space (Real and complex inner-product spaces and their induced length), with inner product linear in the first argument and conjugate-linear in the second, and with the induced length $\mathrm{\parallel }v\mathrm{\parallel }=\sqrt{⟨v,v⟩}$. The definitions below do not require completeness. When the term Hilbert basis is used, $H$ is additionally assumed complete (Hilbert space).

Orthonormal indexed family. Let $I$ be any set. An indexed family $(e_i{)}_{i\in I}$ of vectors of $H$ is orthonormal when

$⟨e_i,e_j⟩={\delta }_{ij}:=\{\begin{array}{ll}1,& i=j,\\ 0,& i\mathrm{\ne }j,\end{array}\text{for all }i,j\in I\mathrm{.}$Because $1\mathrm{\ne }0$ in the scalar field, the family contains no zero vector: $\mathrm{\parallel }{e}_i{\mathrm{\parallel }}^2=⟨e_i,e_i⟩=1$, so $\mathrm{\parallel }{e}_i\mathrm{\parallel }=1$ for every $i$ (Real and complex inner-product spaces and their induced length); and the index map is injective, since $e_i=e_j$ with $i\mathrm{\ne }j$ would give $0=⟨e_i,e_j⟩=1$. Thus an orthonormal family is the same thing as an orthonormal set together with a labelling of it, and in the sequel no choice is hidden in passing between the two descriptions. Orthonormal set. A subset $S\subseteq H$ is orthonormal when every $s\in S$ has $\mathrm{\parallel }s\mathrm{\parallel }=1$ and $⟨s,t⟩=0$ for all distinct $s,t\in S$. For such an $S$ the family $(e_s{)}_{s\in S}$ with $e_s:=s$ is orthonormal in the indexed sense, and the image of an orthonormal indexed family is an orthonormal set. Orthogonal means $⟨s,t⟩=0$ (Orthogonality and the orthogonal complement). Independence and unique coefficients. Let $(e_i{)}_{i\in I}$ be orthonormal, let $F\subseteq I$ be finite, and let scalars $c_i$ satisfy ${\sum }_{i\in F}{c}_i{e}_i=0$. For each fixed $j\in F$, linearity in the first argument gives$0=⟨{\sum }_{i\in F}{c}_i{e}_i,e_j⟩={\sum }_{i\in F}{c}_i⟨e_i,e_j⟩=c_j,$

so every finite subfamily of an orthonormal family is linearly independent. In particular the coefficients in a finite expansion are unique: if ${\sum }_{i\in F}{c}_i{e}_i={\sum }_{i\in F}{d}_i{e}_i$, then applying the computation to $c-d$ gives $c_i=d_i$ for every $i\in F$.

Closed linear span and completeness. The span of an indexed family is the set of all finite linear combinations ${\sum }_{i\in F}{c}_i{e}_i$ with $F\subseteq I$ finite and scalars $c_i$; it is the smallest linear subspace of $H$ containing every $e_i$ (Linear subspace of a vector space). Its closure in the induced norm is the closed linear span of the family, the smallest closed linear subspace of $H$ containing every $e_i$. The family is complete, or is a complete orthonormal system, when its closed linear span is all of $H$. A Hilbert basis, or orthonormal basis, of $H$ is a complete orthonormal family in $H$.

The empty family. The span of the empty family is $\{0\}$, which is already closed, so the empty family is complete exactly when $H=\{0\}$. Thus the zero Hilbert space always has a Hilbert basis, namely the empty one.

Not a Hamel basis, and no order is assumed. A Hilbert basis is a basis only in the sense of closed linear span: for an infinite-dimensional $H$ the vectors of $H$ are in general not finite linear combinations of a Hilbert basis, and the expansion of an arbitrary vector is a norm limit of finite partial sums, not a finite sum. No enumeration, ordering or countability of the index set is part of the definition; the partial sums are indexed by the finite subsets of $I$, ordered by inclusion, and that is the convention used on this page.

Coefficient notation. For orthonormal $(e_i{)}_{i\in I}$ and $x\in H$ the scalars $⟨x,e_i⟩$ are the coefficients of $x$ with respect to the family. They are well defined for every $x$, without any assumption that the family is complete.