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

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The parallelogram law in $L^{2}$

Statement

For $u, v \in L^{2} (μ)$ one has

$∥ u + v ∥_{2}^{2} + ∥ u - v ∥_{2}^{2} = 2 ∥ u ∥_{2}^{2} + 2 ∥ v ∥_{2}^{2} .$

Facts & Assumptions

Given: Classes $u, v \in L^{2} (μ)$ with measurable representatives $f, g$ .

[L1]

The $L^{2}$ norm is well defined on quotient classes (The $L^{p}$ norm descends to the quotient and makes $L^{p}$ a normed space for $1 \leq p \leq \infty$ ).

[L2]

Products of two $L^{2}$ functions lie in $L^{1}$ (Generalized Holder inequality puts products into $L^{r}$ ).

[L3]

The Lebesgue integral is linear on $L^{1}$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

Proof

Proof technique: Expand $∣ f + g ∣^{2} + ∣ f - g ∣^{2}$ pointwise to $2 ∣ f ∣^{2} + 2 ∣ g ∣^{2}$ and integrate.

1.1

Because $f, g \in L^{2} (μ)$ , step [L2] puts $f g$ in $L^{1} (μ)$ , so every term [L2, L3, given, algebra] in the algebraic expansions below is integrable. Pointwise, $∣ f + g ∣^{2} + ∣ f - g ∣^{2} = (f + g)^{2} + (f - g)^{2} = 2 f^{2} + 2 g^{2} .$ Integrating and using [L3] gives $\int ∣ f + g ∣^{2} d μ + \int ∣ f - g ∣^{2} d μ = 2 \int ∣ f ∣^{2} d μ + 2 \int ∣ g ∣^{2} d μ .$

2.1

Rewriting the four integrals as $∥ u + v ∥_{2}^{2}$ , $∥ u - v ∥_{2}^{2}$ , [step 1.1, L1] $∥ u ∥_{2}^{2}$ , and $∥ v ∥_{2}^{2}$ is legitimate by [L1]. That yields the parallelogram identity. ∎

Depends on

Used by

The parallelogram law on two indicator functions in L² Example

Dependency tree · two levels

17 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

John K. Hunter, Measure Theory, Chapter 15 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Section 7B (standard reference, not scraped)

The parallelogram law in L2L2