Georgia Institute of Technology College of Sciences

1. Short background review (mostly covered with handouts).

i. Countable and uncountable sets.

ii. Sequences and series of numbers, Cauchy sequences, liminf and limsup.

iii. Pointwise convergence of sequences and series of functions.

2. Norms and Banach spaces.

i. Definition and properties of norms; examples, especially ℓp.

ii. Convergence in a normed space.

iii. Completeness and Banach spaces.

iv. The uniform norm and uniform convergence.

3. Basic topology in finite- and infinite-dimensional normed spaces.

i. Open sets, closed sets, bounded sets.

ii. Compact sets, with equivalent definitions and comparison of compactness in finite versus infinite dimensions.

4. Lebesgue measure via open and compact sets.

i. Exterior Lebesgue measure.

ii. Lebesgue measure.

iii. Properties of Lebesgue measure, including continuity from above and below.

iv. Example of a non-measurable set.

5. Lebesgue measurable functions.

i. Lebesgue measurable functions.

ii. Almost everywhere convergence and convergence in measure.

6. Lebesgue integral.

i. Definition and basic properties.

ii. Convergence theorems (MCT, Fatou, DCT), relation to the Riemann integral.

iii. Interchange theorems (Fubini and Tonelli).

iv. Absolute continuity and the Fundamental Theorem of Calculus.

7. Lp spaces, including the H¨older and Minkowski inequalities and completeness.

8. Hilbert spaces.

i. Inner products.

ii. Orthogonal projections.

iii. Orthonormal bases.

9. Abstract measures.

i. Radon–Nikodym decomposition with respect to the Lebesgue measure.

ii. Singular measures, absolutely continuous measures, discrete measures.

iii. Analogues for abstract measures of results for Lebesgue measure.

iv. Probability measures.

10. Fourier series and Fourier transforms.

i. Fourier series.

ii. The Fourier transform.