ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 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.

Differentiation on polynomials is unbounded for the supremum norm

Example

Let $P [0, 1]$ be the real polynomial functions on $[0, 1]$ with the supremum norm, and let

$D : P [0, 1] \to P [0, 1], D p : = p^{'} .$

Then $D$ is linear but not bounded for the supremum norm.

Facts & Assumptions

Given: The differentiation operator $D$ on $P [0, 1]$ and the polynomials $p_{n} (x) : = x^{n}$ .

[L1]

A bounded linear operator requires one constant $C$ such that $∥ D p ∥_{\infty} \leq C ∥ p ∥_{\infty}$ for every polynomial $p$ (A bounded linear operator between normed spaces).

Verification

technique · direct

1.1

Differentiation is linear on polynomials. For each $n \geq 1$ , $∥ p_{n} ∥_{\infty} = 1$ on $[0, 1]$ , while $D p_{n} (x) = n x^{n - 1}$ and therefore $∥ D p_{n} ∥_{\infty} = n$ .

givenalgebra

2.1

If $D$ were bounded, [L1] would give a constant $C$ with $n = ∥ D p_{n} ∥_{\infty} \leq C ∥ p_{n} ∥_{\infty} = C$ for every $n \geq 1$ , impossible. Hence $D$ is unbounded.

step 1.1L1assume-contradischarge-contradiction∎

Depends on

A bounded linear operator between normed spaces

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)
Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)