False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-31

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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.
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FALSE: $L^{1 / 2}$ with its $p$ -functional is a normed space

Statement

The space $L^{1 / 2} (μ)$ with the functional $∥ \cdot ∥_{1 / 2}$ is a normed space.

Facts & Assumptions

Given: The explicit half-norm triangle-inequality failure.

[L1]

The previous counterexample exhibits functions with $∥ f + g ∥_{1 / 2} > ∥ f ∥_{1 / 2} + ∥ g ∥_{1 / 2}$ (The half-norm fails the triangle inequality on two indicators).

[L2]

A normed space requires a norm, hence the triangle inequality (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).

Refutation

Proof technique: Refute with the explicit $1 / 2$ -triangle-inequality failure on two indicators.

1.1

The functions from [L1] violate the triangle inequality required by [L2].

L1L2

2.1

Therefore $L^{1 / 2} (μ)$ with $∥ \cdot ∥_{1 / 2}$ is not a normed space.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Theorem 8.16 (standard reference, not scraped)

FALSE: L1/2L1/2 with its pp-functional is a normed space