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

Definition

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

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

Remarks

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

Used by

Dependency tree · two levels

6 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

Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)