TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-09

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Dominated extension conditional on the relative principle

Statement

Assume HB. Let $X$ be a real vector space, $M \leq X$ a real linear subspace, $p : X \to R$ sublinear, and $g : M \to R$ real linear with $g (m) \leq p (m)$ for every $m \in M$ . There is a real linear extension $F : X \to R$ satisfying $- p (- x) \leq F (x) \leq p (x) (x \in X) .$ No topology, closedness, or completeness is required.

Facts & Assumptions

[F1]

HB asserts a dominated real linear extension for every such quadruple $(X, M, p, g)$ (The real dominated-extension principle as an additional hypothesis over ZF).

Proof

Given: HB and $X, M, p, g$ as in the statement.

1.1givenF1

All four objects have the types required by HB, and the given inequality holds for every $m \in M$ . Applying HB to this quadruple yields real linear $F$ with $F ∣_{M} = g$ and $F (x) \leq p (x)$ for every $x \in X$ .

2.1step 1.1algebra∎

For each $x \in X$ , also $- x \in X$ , so $F (- x) \leq p (- x)$ . Since $F (- x) = - F (x)$ , multiplication by $- 1$ gives $- p (- x) \leq F (x)$ . Together with the upper bound this proves the claim. At zero, $p (0) = F (0) = 0$ , so both inequalities are equalities.

Source notes

Brezis Theorem 1.1, p.1; Teschl Theorem 4.13 and following lower-bound observation, pp.112–113.

Depends on

The real dominated-extension principle as an additional hypothesis over ZF