Syllabus for the Analysis (M805, M806, M807)
Candidacy Exam

Approved by the graduate committee on 5/16/99

Note: Students must choose to be examined in two of the following areas when registering to take the exam:

Real analysis (M805)

Functional Analaysis (M806)

Complex Variables (M807)

Real Analysis Component (M805)

• Properties of Lebesgue outer measure and Lebesgue measurable sets in .
• Properties of Lebesgue measurable functions. Concepts of convergence almost everywhere and convergence in measure.
• Properties of Lebesgue integrals and various convergence theorems. Relation between Lebesgue integral and Riemann-Stieltjes integrals.
• Repeated integration. Fubini's and Tonelli's Theorems.
• Differentiation. Vitali's covering lemma and its applications. Absolutely continuous and singular functions.
• Spaces.

References:

1. Measure and Integral by R. Wheeden and A. Zygmund
2. Real Analysis by H. Royden

Further References:

3. Theory of Functions of Real Variables by I. Natanson

4. Theory of Functions by E.C. Titchmarsh

Functional Analysis Component (M806)

• Normed linear spaces; Banach spaces and Hilbert spaces
• Linear operators
• Linear functionals; Hahn Banach theorem
• Open mapping theorem; closed graph theorem
• Uniform boundedness principle
• Compact operators Hilbert-Schmidt theorem and Fredholm alternative
• Weak topology

References

1. Kreyszig, ``Introduction to Functional Analysis''

Complex Variables Component (M807)

• Limits, continuity and uniform convergence;
• Differentiation, analytic functions and elementary properties of conformality;
• Mobius transformation;
• Complex integrals, Cauchy integral theorem;
• Cauchy integral formula and its implications;
• Power series, uniform convergence;
• Laurent series; singular points;
• The residue theorem and its implications.

References:

1. L. Alfors, ``Complex Analysis"
2. J. Conway, ``Functions of a Complex Variable".

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