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.
Elements of are equivalence classes, so pointwise statements require a representative
Elements of The space as the quotient by null functions are classes , not chosen functions. So formulas such as "" or "" are not meaningful until a representative is named. Quantities invariant under changing on a null set, such as , are meaningful on the quotient; pointwise assertions are not.
Depends on
Used by
- Point evaluation at 0 is not well defined on Lᵖ[0,1] Counterexample
- For 1 ≤ p < ∞, every Lᵖ(μ) class has a sigma-finite essential support Lemma
- The functional Λ_g has norm ‖g‖_q; for q=∞ assume μ is semifinite Proposition
- Every g∈ L^q(μ) defines a bounded linear functional on Lᵖ(μ) Theorem
- Simple functions are dense in L^∞(μ) in the essential-supremum norm Theorem
- Simple functions with finite-measure support are dense in Lᵖ(μ) for 1 ≤ p < ∞ Theorem
Dependency tree · two levels
5 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 7B (standard reference, not scraped)