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Every finite-dimensional real or complex inner product space has an orthonormal basis
Statement
Every finite-dimensional real or complex inner product space has an orthonormal basis. In dimension zero, this is the empty basis.
Facts & Assumptions
Given: A finite-dimensional inner product space .
A finite-dimensional vector space has a finite basis, with the empty list serving when (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).
Gram–Schmidt converts every finite independent list into an orthonormal list with the same span (Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans).
Proof
Choose a finite basis of using [L1].
Apply [L2]. The resulting orthonormal list has the same span as the basis, namely , and therefore is an orthonormal basis. This also covers .
Depends on
Used by
- 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
- Hilbert exterior powers and induced operators Definition
- Mean curvature vector Definition
- The oriented unit volume form Definition
- Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
- Local separable trace-class determinant construction Lemma
- Positive square root of a compact positive operator Lemma
- Orthogonal complements of subbundles are smooth subbundles Proposition
- Ricci decomposition of the Riemann tensor in dimension at least three Proposition
- Cut locus of a point has riemannian volume zero Theorem
- Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition Theorem
- Finite-dimensional Riesz representation: every functional is uniquely v↦⟨ v,w⟩ Theorem
- For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent Theorem
- Interior product is the adjoint of exterior multiplication by a vector Theorem
- Singular value decomposition for compact operators Theorem
- Spectral theorem for compact self adjoint operators Theorem
- The Gram formula gives a well-defined positive-definite inner product on exterior powers, and ‖v₁∧⋯∧ vₖ‖² is the Gram determinant Theorem
- Trace of a positive operator is the sum of its eigenvalues Theorem
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)