Every finite-dimensional real or complex inner product space has an orthonormal basis

Depends on

• Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans
• Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis

Used by

• Exact Gaussian randomized range finder Algorithm
• Reorthogonalized randomized power subspace iteration Algorithm
• Exact-arithmetic CG after grounding or a Neumann gauge Corollary
• Orthogonal and unitary operators form groups, and their determinants have modulus one Corollary
• Orthonormal eigenbasis for a compact self adjoint operator Corollary
• Polar integration may discard the cut locus Corollary
• Zero scalar curvature does not imply flatness Counterexample
• Controlled Unitary Access Definition
• Dual Coxeter number for the normalized invariant form Definition
• Hilbert exterior powers and induced operators Definition
• Mean curvature vector Definition
• Partial trace over the first tensor factor Definition
• Pure-State Entanglement Across a Tensor Cut Definition
• Quantum acceptance probability for a finite promise computation Definition
• Row leverage scores of a finite-dimensional subspace Definition
• The normalized irreducible matrix coefficient family Definition
• The oriented unit volume form Definition
• The unitary dual of a compact group Definition
• A positive activation mean creates a high-variance common initialization mode Example
• Complex finite-basis PC-VAR optimization on Grassmann Example
• Distance hessian and laplacian in space forms Example
• Link edge lengths versus dihedral mirror angles in type I₂(m) Example
• Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
• Toponogov comparison on a round sphere Example
• A finite isometry extends after finite ancilla padding Lemma
• A finite POVM defines a classical-output quantum channel Lemma
• A full-column-rank real matrix has an economy QR factorization Lemma
• A positive Choi matrix yields finite Kraus operators Lemma
• A separable type I factor is a multiple of an irreducible representation Lemma
• Adversary progress at initial and bounded-error final states Lemma
• Agreement on pure states implies complete agreement Lemma
• Amplitude-amplification invariant plane Lemma
• An orthogonal finite ensemble block sum is a density operator Lemma
• Bounded density and finite-vector transitivity for C*-representations Lemma
• Coherent normal form for finite quantum query algorithms Lemma
• Complement and degradation maps of the erasure channel Lemma
• Complete positivity implies Choi positivity Lemma
• Detectable operators form a subspace and quantum distance is well defined Lemma
• Direct sums and tensor products of finite-dimensional unitary representations Lemma
• Euclidean balls are bounded C-one domains with radial outward normal Lemma

…and 86 more results.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Sheldon Axler, Linear Algebra Done Right, 4th ed., result 6.35 (standard reference, not scraped)