Lebesgue Spaces

This discusses the Lebesgue space $L^{p}$. For measure or Lebesgue integration itself, refer to the Measure Theory category.

Lebesgue Spaces

• $L^{p}$ spaces, Lebesgue spaces
  • Why define the inner product of functions as a definite integral
  • $L^{1}$ space
  • $L^{p}$ convergence
• $L^{\infty}$ space
• Weighted $L^{p}$ spaces
• $L_{\text{loc}}^{1}$ space

Basic Properties

• Cauchy-Schwarz inequality
• Hölder’s inequality
  • Generalized Hölder’s inequality
  • Reverse Hölder’s inequality
  • A sufficient condition for a function to be in $L^{p}$
• Minkowski’s inequality
  • Reverse Minkowski’s inequality
• Interpolation inequality
• Young’s theorem
• Embedding theorems for $L^{p}$ spaces

Uniform Convexity

• Proof of Clarkson’s inequalities
• Proof that $L^{p}$ spaces are uniformly convex and reflexive

Norm Dual

• Linear functionals on $L^{p}$ spaces
• Riesz representation theorem for $L^{p}$ spaces

$L^{2}$ Space

• $L^{2}$ space
  • Relationship between $L^{1}$ and $L^{2}$ spaces
• Orthogonal functions and orthogonal sets
• Orthonormal bases, completeness, orthonormal sets
• Bessel’s inequality
• Translation, modulation, dilation
  • Inverse operators
  • Commutation relations
• 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)