DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators

Let $X$ and $Y$ be normed spaces over the same scalar field. Write

$B (X, Y) : = {T : X \to Y : T is a bounded linear operator} .$

This is a subspace of the vector space $L (X, Y)$ of The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication, with the same pointwise operations:

$(S + T) (x) : = S x + T x, (λ T) (x) : = λ T x .$

If $X = Y$ , write

$B (X) : = B (X, X) .$

Boundedness is preserved by the two pointwise operations, because the triangle inequality and absolute homogeneity combine the corresponding bounds.
The operator norm of The operator norm as the least bound and as the unit-sphere or unit-ball supremum is defined on $B (X, Y)$ , not on all of $L (X, Y)$ .

Depends on