Lebesgue Spaces
This discusses the Lebesgue space $L^{p}$. For measure or Lebesgue integration itself, refer to the Measure Theory category.
Lebesgue Spaces
Basic Properties
- Cauchy-Schwarz inequality
- Hölder’s inequality
- Minkowski’s inequality
- Interpolation inequality
- Young’s theorem
- Embedding theorems for $L^{p}$ spaces
Uniform Convexity
Norm Dual
$L^{2}$ Space
- $L^{2}$ space
- Orthogonal functions and orthogonal sets
- Orthonormal bases, completeness, orthonormal sets
- Bessel’s inequality
- Translation, modulation, dilation
- Fourier transform as an operator
Sturm-Liouville Form
- Sturm-Liouville differential equations
- Eigenvalues and eigenfunctions in S-L problems
- Orthogonality of solutions in regular Sturm-Liouville problems
Key References
- Capinski, Measure, Integral and Probability (1999)
- Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications (2nd Edition, 1999)
- Robert A. Adams and John J. F. Foutnier, Sobolev Space (2nd Edition, 2003)
- Ole Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (2010)
All posts
- L1 Space
- L2 Space
- Cauchy-Schwarz Inequality in Lebesgue Spaces
- Reason for Defining the Inner Product of Functions via Definite Integration
- Lp Spaces, Lebesgue Spaces
- Proof of Hölder's Inequality in Lebesgue Spaces
- Proof of Minkowski's Inequality in Lebesgue Spaces
- Orthogonal Functions and Orthogonal Sets
- Bessel's Inequality in L2 Space
- Complete Orthonormal Basis and Complete Orthonormal Set
- Generalized Hölder's Inequality, Corollaries of Hölder's Inequality
- The Relationship between L1 Space and L2 Space
- Fundamental Theorem of Algebra
- Linear Functionals on Lp Spaces
- Interpolation Inequalities in Lebesgue Spaces
- Proof of Clarkson's Inequalities
- Holder's Inequality
- Minkowski's Inequality
- The Riesz Representation Theorem for Lp Spaces
- Proving that Lp Spaces are Uniformly Convex and Reflexive
- Lp Convergence
- Translation in L2 Spaces: Translations, Modulations, Dilations
- Translation, Modulation, and Dilation Operators in L2 Space
- Translation, Modulation, and Dilation Commutation Relations in L2 Spaces
- Operators as Fourier Transforms
- Locally Integrable Function
- Sturm-Liouville Differential Equation
- Eigenvalues and Eigenfunctions in S-L Problems
- Orthogonality of Solutions to the Regular Sturm-Liouville Problem
- Weighted Lp Spaces
- Inverse Holder's Inequality: A Sufficient Condition for Lp Functions
- L infinity space
- Embedding Theorems in Lp Spaces