Orthonormal basis

An orthonormal basis of a Hilbert space is an orthonormal set whose linear span has the whole space as its closure. This completeness of an orthonormal system allows every vector to be expanded in the basis vectors, with convergence in norm. In finite dimension it is also a basis in the finite-linear-combination sense.

An orthonormal set is complete in a Hilbert space when the closure of its linear span is the whole space. Equivalently, the only vector orthogonal to every member is zero: project any vector onto the closed span, and its remaining orthogonal component must vanish. This is completeness of an expansion system, distinct from completeness of the ambient metric space.

Coordinates relative to an origin $\mathbf{o}$ and an orthonormal basis ${\mathbf{e}}_1,\dots ,{\mathbf{e}}_n$ preserve Euclidean distances: if $\mathbf{x}-\mathbf{y}={\sum }_i{d}_i{\mathbf{e}}_i$, then $\Vert \mathbf{x}-\mathbf{y}{\Vert }^2={\sum }_i{d}_i^2$. This matters when finding the eccentricity of a conic section in an oblique plane: arbitrary projected coordinates need not preserve the ratio of its semi-major axis to its semi-minor axis.

A complete orthonormal basis in a Hilbert space resolves the identity as ${\sum }_j|j⟩⟨j|=I$. In infinite dimension the partial sums converge strongly: their action on each fixed vector converges in norm. Generalized position and momentum eigenstates instead give distributional integral resolutions, such as $\int dx|x⟩⟨x|=I$, with their normalization interpreted through the Dirac delta function.

For a real orthonormal basis $u_1,\dots ,u_n$, each vector has expansion $x={\sum }_j{u}_j(u_j^{T}x)$. Since this holds for every $x$, the sum of the rank-one orthogonal projection matrices $u_j{u}_j^T$ is the identity matrix. Their pairwise products vanish for distinct basis vectors. For a complex inner product space, replace the transpose by the conjugate transpose.