Syllabus/achievement requirements

As a main reference we use

[D] B. Driver "Analysis tools with examples"

For the last part on harmonic analysis we use

[W] T. Wolff "Lectures on harmonic analysis"

Here are some other options:

[P] G. Pedersen "Analysis now"

[F] G. Folland "Real analysis"

[MDW] J. McDonald, N. Weiss "A course in real analysis"

[KF] A. Kolmogorov, S. Fomin "Elements of the theory of functions and functional analysis"

Syllabus:

Fundamental principles of functional analysis ([D]: 25.1-25.3)

Product measures, Fubini-Tonelli theorem ([D]: 18.4, 20.1, 32.9)

Modes of convergence, Egorov theorem ([D]: 21.2)

Radon-Nikodym derivative and duals of Lp-spaces ([D]: 24.1)

Complex and signed measures ([D]: 24.2-24.3)

Riesz-Markov theorem ([D]: 28.2, 31.5)

Lebesgue differentiation theorem ([D]: 30.1-30.2)

Absolutely continuous functions ([D]: 30.4)

Haar measure ([F]: 11.1 or this)

Convolution, Fourier transform, Plancherel theorem ([W]: pp. 1-8, 13-18)