Real Analysis

Undergraduate Analysis

• References
  • Rudin: Chapters 2, 3, 4, 5, 7
  • Folland: Section 0.6

• Continuity and differentiation of functions $f: {\mathbf{R}}\to {\mathbf{R}}$

• 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 ${\mathbf{R}}^n$ 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 $( L^2([0, 1]), {\left\lVert {{-}} \right\rVert}_2 )$