= Unitary extension of a finite-dimensional isometry
{title2=$\langle u_i,u_j\rangle=\delta_{ij}$}
= Unitary extension
{synonym}
A <linear isometry> from a subspace of a finite-dimensional <Hilbert space> into the same ambient space extends to a <unitary operator>. Complete an <orthonormal basis> of the input subspace and its isometric image to full <orthonormal bases>, then map the first to the second. The equal complement dimensions make this possible. Orthogonality of the specified image columns, not normalization alone, is essential. In an infinite-dimensional space arbitrary isometries need not be surjective; the finite-dimensional same-space hypothesis cannot be dropped.
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