Real Analysis

Undergraduate Analysis

  • References
    • Rudin: Chapters 2, 3, 4, 5, 7
    • Folland: Section 0.6
  • Continuity and differentiation of functions \(f: {\mathbb{R}}\to {\mathbb{R}}\)

  • Metric spaces

  • Compactness in analysis

  • Sequences and series

  • Uniform convergence

  • Uniform continuity

  • Taylor’s theorem

  • Weierstrass Approximation Theorem

Measure and Integration

  • References:
    • Folland, Chapters 1, 2
    • Stein and Shakarchi, Chapters 1, 2, 6
  • Measures on \({\mathbb{R}}^n\) and on \(\sigma{\hbox{-}}\)algebras

  • Measurable functions

  • Integrable functions

  • Convergence theorems:

    • Fatou’s Lemma
    • The Monotone Convergence Theorem
    • The Dominated Convergence Theorem
  • Egorov’s theorem

  • Notions of convergence:

    • Uniform
    • Pointwise
    • Almost everywhere
    • In norm
  • Fubini and Tonelli theorems

Function Spaces

  • References:
    • Folland: Sections 5.2, 5.5, 6.2;
    • Stein and Shakarchi: Chapter 4.
  • The Banach spaces \(L^1\) and \(L^\infty\):

  • Completeness

  • Convolutions

    • Approximations to the identity
  • Linear functionals and

    • \(L^\infty\) as the dual of \(L^1\)

Hilbert space and \(L^2\) spaces:

  • References:
    • Folland: Sections 5.2, 5.5, 6.2;
    • Stein and Shakarchi: Chapter 4.
  • Schwarz inequality

  • Orthogonality

  • Linear functionals

  • The Riesz representation theorem

  • Bessel’s inequality

    • Orthonormal bases
    • Parseval’s identity
  • Trigonometric series:

    • Trigonometric polynomials are dense in \(( C([0, 1]), {\left\lVert {{-}} \right\rVert}_\infty)\)
    • Trigonometric polynomials are dense in \(( L^2([0, 1]), {\left\lVert {{-}} \right\rVert}_2 )\)
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