The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions

Depends on

• The essential supremum of a measurable function with respect to a measure

Used by

• A positive reaction term restores coercivity without Poincar'e Corollary
• One-dimensional W^1,p functions have unique absolutely continuous representatives Corollary
• Positive, negative, and truncated Sobolev functions Corollary
• Sobolev maxima and minima form a lattice Corollary
• The zero-countable / infinity-cocountable measure space breaks the p=1 endpoint of duality Counterexample
• Measurable Beltrami coefficients and measurable conformal structures Definition
• Sublinear operators and weak or strong type (p,q) bounds Definition
• The space Lᵖ(μ) as the quotient by null functions Definition
• Uniformly elliptic divergence-form operators and their sesquilinear forms Definition
• A clipped affine function keeps its zero region Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Sharp Sobolev threshold for a radial power Example
• Bounded compact data give an everywhere finite Newtonian potential Lemma
• The elliptic form is well defined and bounded on H¹ Lemma
• Null functions form a linear subspace and are exactly the zero-seminorm class Proposition
• The centered maximal operator is bounded on L^∞ Proposition
• Chain rule for a C¹ function with bounded derivative Theorem
• Chain rule for globally Lipschitz scalar maps of Sobolev functions Theorem
• Finite-measure Lʳ includes into Lᵖ for p < r Theorem
• Garding's inequality for a divergence-form elliptic operator Theorem
• Generalized Holder inequality puts products into Lʳ Theorem
• Holder's inequality for integrals, including the endpoint cases Theorem
• L^∞[0,1] is not separable Theorem
• Lax--Milgram solvability for coercive divergence-form equations Theorem
• Lᵖ and L^∞ are vector spaces for p ≥ 1 Theorem
• Lᵖ norms converge to the essential supremum for essentially bounded Lʳ functions Theorem
• Minkowski's inequality for integrals, including p = ∞ Theorem
• Newtonian potentials solve the distributional Poisson equation Theorem
• Raikov: compact-open and weak star topologies agree on normalized positive type functions Theorem
• Simple functions are dense in L^∞(μ) in the essential-supremum norm Theorem

Dependency tree · two levels

2 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, Section 7.1 (standard reference, not scraped)
• Sheldon Axler, Measure, Integration & Real Analysis, Section 7A (standard reference, not scraped)