Alphabeta MathHow 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 function space for
Definition
Let be a measure space and let be a real number with . Write for the measurable functions . For define the extended-valued -functional
The finite power uses Real powers for positive bases, with the zero-base positive-exponent convention, while the second clause avoids applying real powers to the extended value . The integral is the nonnegative Lebesgue integral of The nonnegative Lebesgue integral.
The class
is the measurable-function space used on this page before passing to almost-everywhere equivalence classes.
For , the pointwise operations make a real vector space by and are vector spaces for .
Depends on
Used by
- A Cauchy sequence in calligraphic Lᵖ can converge to two distinct functions Counterexample
- The space Lᵖ(μ) as the quotient by null functions Definition
- x⁻ᵃ on (0,1) and on (1,∞) calibrates Lᵖ membership Example
- Null functions form a linear subspace and are exactly the zero-seminorm class Proposition
- Equality in Holder's inequality for 1 < p < ∞ Theorem
- Finite-measure Lʳ includes into Lᵖ for p < r Theorem
- Generalized Holder inequality puts products into Lʳ Theorem
- Holder's inequality for integrals, including the endpoint cases Theorem
- Lᵖ and L^∞ are vector spaces for p ≥ 1 Theorem
- Lᵖ norms converge to the essential supremum for essentially bounded Lʳ functions Theorem
- Lyapunov interpolation inequality for Lᵖ norms Theorem
- Minkowski's inequality for integrals, including p = ∞ Theorem
- The p-power triangle inequality for nonnegative functions when 0 < p < 1 Theorem
Dependency tree · two levels
9 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, Section 7A (standard reference, not scraped)
- John K. Hunter, Measure Theory, Section 7.1 (standard reference, not scraped)