Real Analysis

Recurring questions

Chapters

1. Undergraduate analysis – sequences, continuity, differentiability, compactness.

2. Measure – outer measure, measurability, Littlewood’s principles.

3. Integration – the convergence theorems and \(L^1\).

4. Fubini and Tonelli – product measures and interchanging iterated integrals.

5. $L^p$ spaces – the inequalities, duality, density.

6. Fourier – convolution, the transform, approximate identities.

7. Functional analysis – Banach and Hilbert spaces, operators, and duality.

Three pages collect material from several chapters:

• Review sheet – the statements of the chapters’ theorems, on one page.

• Counterexamples – grouped by the statement each refutes.

• Inequalities – grouped by the quantity each bounds.

Further pages: appendices, resources, and workshops.

Acknowledgements

Thanks to Peter Woolfitt for supplying many solutions and checking many proofs in problem sessions.

Topics and references

Undergraduate analysis

• References:

  • [Rudi76]

  • [Foll13]

• Continuity and differentiation of functions \(f: \RR \to \RR\)

• Metric spaces

• Compactness in analysis

• Sequences and series

• Uniform convergence

• Uniform continuity

• Taylor’s theorem

• Weierstrass approximation theorem

Measure and integration

• References:

  • [Foll13]

  • [StSh05]

• Measures on \(\RR^n\) and on \(\sigma\dash\)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:

  • [Foll13]

  • [StSh05]

• The Banach spaces \(L^1\) and \(L^\infty\)

  • Completeness

• Convolutions

  • Approximations to the identity

• Linear functionals

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

Hilbert spaces and \(L^2\)

• References:

  • [Foll13]

  • [StSh05]

• Schwarz inequality

• Orthogonality

• Linear functionals

• The Riesz representation theorem

• Bessel’s inequality

  • Orthonormal bases

  • Parseval’s identity

• Trigonometric series:

  • Trigonometric polynomials are uniformly dense in the continuous functions on \([0,1]\) with matching endpoint values.

  • Trigonometric polynomials are dense in \(( L^2([0, 1]), \norm{\wait}_2 )\)