The space $L^p(\mu )$ as the quotient by null functions

Depends on

• The function space $\mathcal{L}^p(\mu)$ for $0 < p < \infty$
• The space $L^\infty(\mu)$ of essentially bounded measurable functions
• The null subspace of measurable functions that vanish almost everywhere
• The quotient vector space $V/W$ and its canonical projection

Used by

• Bounded Sobolev sequences have strongly convergent subsequences with the weak limit as limit Corollary
• Compact and discrete transforms are the two extreme Plancherel cases Corollary
• Conditional lp contraction Corollary
• Finite propagation for scalar conservation laws Corollary
• Global L¹ contraction from the local estimate Corollary
• Mass conservation for compactly supported entropy solutions Corollary
• Non-invertible elliptic shifts form a discrete set in the self-adjoint case Corollary
• Poincare-Wirtinger on bounded convex domains by the direct pairwise argument Corollary
• Positive, negative, and truncated Sobolev functions Corollary
• Scaled Caccioppoli inequality on concentric balls Corollary
• Sobolev maxima and minima form a lattice Corollary
• Strong convergence of subcritical powers Corollary
• Strong maximum principle for weak elliptic solutions Corollary
• Subcritical compactness for W^1,p₀ on arbitrary bounded open sets Corollary
• The L^∞ maximum bound for entropy solutions Corollary
• The Nyquist no-aliasing condition Corollary
• The Poincare constant is the reciprocal square root of the first Dirichlet eigenvalue Corollary
• The Sobolev inequality for zero-boundary Sobolev closures on open sets Corollary
• The Sobolev space W^k,p is an algebra above the critical index Corollary
• Uniqueness, comparison and order preservation of entropy solutions Corollary
• Weak differentiation has a closed graph on its natural domains Corollary
• Weak H¹ convergence plus compactness gives strong L² convergence Corollary
• A gradient-only Poincare estimate needs normalisation Counterexample
• A jump boundary datum is outside the trace range for p≥2 Counterexample
• A large adverse zero-order term destroys Dirichlet coercivity Counterexample
• A mild heat solution need not be classical at the initial time Counterexample
• A mild solution need not be classical Counterexample
• A nonconvex gradient energy can lose weak lower semicontinuity Counterexample
• A norm need not satisfy the parallelogram law Counterexample
• A normal operator need not have any eigenvectors Counterexample
• A normalised cube indicator is not an H¹ atom Counterexample
• A strongly continuous unitary group need not be norm continuous Counterexample
• A symmetric closed operator that is not self-adjoint Counterexample
• Arbitrary L² boundary data need not have an H¹ lifting Counterexample
• Backward heat amplifies small high-frequency errors Counterexample
• Boundary point values are not a function of the interior Lᵖ class Counterexample
• Continuous calculus does not contain discontinuous spectral projections Counterexample
• Continuous functional calculus cannot produce every spectral projection Counterexample
• Critical traces fail compactness under boundary dilation Counterexample
• Elliptic eigenvalues need not be simple Counterexample

…and 233 more results.

Dependency tree · two levels

11 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, Definition 7.17 (standard reference, not scraped)
• John K. Hunter, Measure Theory, Section 7.4 (standard reference, not scraped)