DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

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The complex $L^{2}$ pairing on equivalence classes

Definition

For $[f], [g] \in L^{2} (μ; C)$ , the proposed pairing is $⟨ [f], [g] ⟩ : = \int_{X} f \overline{g} d μ .$ The integral is computed from measurable representatives. By Complex Holder, Minkowski, and the quotient norm, $\overline{g} \in L^{2}$ and $\int ∣ f \overline{g} ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2} < \infty$ , so this representative expression is defined. The measurable and integration conventions are those of Complex Lp classes and Euclidean test-function conventions. This fixes the first-variable-linear convention. The following theorem, named in justified_by, establishes class invariance and the inner-product axioms; the present definition does not assume that obligation.

Depends on

Used by

Dependency tree · two levels

26 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)

The complex L2L2 pairing on equivalence classes