DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||

Definition

Let $T : V \to W$ be a linear map between finite-dimensional real or complex inner product spaces. If $V = 0$ , define $∥ T ∥ : = 0$ . If $V \neq 0$ , define the operator norm of $T$ by

$∥ T ∥ : = \max {∥ T v ∥ : ∥ v ∥ = 1} .$

The next theorem proves that this maximum exists and equals the largest singular value.

Depends on

The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product

Dependency tree · 2 levels

Sources

Sheldon Axler, Linear Algebra Done Right, fourth edition