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Book 4A

Function Spaces and Sobolev Spaces

Analysis in infinite dimensions

Banach and Hilbert spaces, duality, weak compactness, compact operators and spectra, distributions, and the Sobolev and Hölder spaces in which geometric PDE are solved.

  • Main companion (first pass)Introductory Functional Analysis with Applications · Erwin Kreyszig
  • Main companionFunctional Analysis, Sobolev Spaces and PDE · Haim Brezis
In preparation

Contents

  1. 1
    Banach Spaces and Bounded Operators

    Normed spaces, operator norms, and why the unit ball is compact only in finite dimensions.

    In preparation

  2. 2
    Baire Category and Its Consequences

    Uniform boundedness, open mapping and closed graph, and why some numerical methods diverge.

    In preparation

  3. 3
    Hahn–Banach and Duality

    Separating convex sets, dual spaces, and the dual of Lᵖ.

    In preparation

  4. 4
    Hilbert Spaces and Lax–Milgram

    Projection, Riesz representation, orthonormal bases, and the existence theorem for weak solutions.

    In preparation

  5. 5
    The Fourier Transform

    Plancherel, Gaussians, uncertainty, and solving the heat equation by frequencies.

    In preparation

  6. 6
    Weak Convergence and the Direct Method

    Recovering compactness weakly, and minimising energies.

    In preparation

  7. 7
    Compact Operators and Spectra

    The spectral theorem for compact self-adjoint operators and the eigenvalues of the Laplacian.

    In preparation

  8. 8
    Distributions and Weak Derivatives

    Derivatives of functions that are not differentiable, and fundamental solutions.

    In preparation

  9. 9
    Sobolev Spaces

    Functions with derivatives in Lᵖ, approximation, and the Poincaré inequality.

    In preparation

  10. 10
    Sobolev Embeddings and Critical Exponents

    Scaling, embeddings, Rellich compactness, and how compactness fails at the critical exponent.

    In preparation

  11. 11
    Hölder Spaces

    Measuring roughness, and why Hölder spaces are the right setting for regularity.

    In preparation

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