DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces

Definition

A linear map $T : V \to W$ (Linear map between vector spaces over the same field) between inner product spaces (Real and complex inner product spaces, with the inner product linear in the first argument) is a linear isometry if

$∥ T v ∥ = ∥ v ∥$

for every $v \in V$ , where the norm is the induced norm The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product. An invertible linear isometry from a real finite-dimensional inner product space to itself is an orthogonal operator; over $C$ it is a unitary operator.

Equivalently, once the finite-dimensional characterisation is proved, orthogonal and unitary operators are the endomorphisms satisfying $T^{*} T = T T^{*} = I$ .

Depends on

Used by

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., definitions 7.44 and 7.51 (standard reference, not scraped)