TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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 $L^{p}$ norm descends to the quotient and makes $L^{p}$ a normed space for $1 \leq p \leq \infty$

Statement

Let $(X, A, μ)$ be a measure space.

For $1 \leq p < \infty$ , the rule $∥ [f] ∥_{p} : = ∥ f ∥_{p} ([f] \in L^{p} (μ))$ is well defined on the quotient classes.
For $p = \infty$ , the rule $∥ [f] ∥_{\infty} : = ∥ f ∥_{\infty} ([f] \in L^{\infty} (μ))$ is well defined.
In either case, with the quotient vector-space operations of Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$ , the resulting pair $(L^{p} (μ), ∥ \cdot ∥_{p})$ is a normed space in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms.

Facts & Assumptions

Given: A measure space $(X, A, μ)$ and an exponent $1 \leq p \leq \infty$ .

[L1]

The quotient spaces $L^{p} (μ)$ are those of The space $L^{p} (μ)$ as the quotient by null functions.

[L2]

The null representatives are exactly the zero-seminorm class (Null functions form a linear subspace and are exactly the zero-seminorm class).

[L3]

Minkowski's inequality supplies the triangle inequality for the representative seminorms (Minkowski's inequality for integrals, including $p = \infty$ ).

[L4]

The essential supremum is an attained essential bound (The essential supremum is attained as the least essential bound).

[L5]

The quotient operations are well defined and produce a vector space (Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$ ).

[L6]

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

Proof

Proof technique: Use the previous proposition to identify the null functions as the kernel of the seminorm. Hence the seminorm is constant on cosets and separates points on the quotient, while Minkowski and homogeneity descend from representatives.

1.1

Suppose first $1 \leq p < \infty$ and $f - g \in N_{p} (μ)$ . Then $∥ f - g ∥_{p} = 0$ . Applying Minkowski twice yields [L2, L3] $∥ f ∥_{p} \leq ∥ g ∥_{p} + ∥ f - g ∥_{p} = ∥ g ∥_{p}, ∥ g ∥_{p} \leq ∥ f ∥_{p} + ∥ f - g ∥_{p} = ∥ f ∥_{p},$ so $∥ f ∥_{p} = ∥ g ∥_{p}$ . Thus $∥ [f] ∥_{p} : = ∥ f ∥_{p}$ is well defined.

1.2

For $p = \infty$ , if $f - g \in N_{\infty} (μ)$ then $f = g$ almost everywhere. Any essential bound for $f$ is therefore an essential bound for $g$ and conversely, so $∥ f ∥_{\infty} = ∥ g ∥_{\infty}$ . Thus $∥ [f] ∥_{\infty} : = ∥ f ∥_{\infty}$ is well defined.

L2L4

1.3

For separation, $∥ [f] ∥_{p} = 0$ means $∥ f ∥_{p} = 0$ , and then [L2] gives $f \in N_{p} (μ)$ or $N_{\infty} (μ)$ . Hence $[f] = [0]$ . The converse is immediate because the zero representative has norm $0$ .

2.1

By [L1] and [L5], each quotient $L^{p} (μ)$ is already a real vector space. The representative functionals are homogeneous, and [L3] supplies the triangle inequality on representatives; steps 1.1 and 1.2 show that these formulas depend only on the class, so homogeneity and triangle inequality descend to the quotient.

step 1.1step 1.2L1L3L5

3.1

Steps 2.1 and 1.3 verify the three norm axioms named in [L6]. Therefore $(L^{p} (μ), ∥ \cdot ∥_{p})$ is a normed space for every $1 \leq p \leq \infty$ .

step 2.1step 1.3L6∎

Depends on

Used by

Convergence in Lᵖ implies convergence in measure Corollary
Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞ Theorem
The parallelogram law in L² Theorem

Dependency tree · two levels

34 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

Sheldon Axler, Measure, Integration & Real Analysis, Definition 7.17 and Theorem 7.18 (standard reference, not scraped)
John K. Hunter, Measure Theory, Sections 7.1 and 7.4 (standard reference, not scraped)

The LpLp norm descends to the quotient and makes LpLp a normed space for 1≤p≤∞1≤p≤∞