# Real 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 $$( L^2([0, 1]), {\left\lVert {{-}} \right\rVert}_2 )$$