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Analysis

While studying for the analysis qualifying exam, I typed up solutions for the problems I worked on. Most solutions are taken from the solutions maintained by Adam Lott (see below), but rewritten into my own words. Towards the end, I began writing more original solutions.

I passed with the Spring 2021 exam.

Collaborators

Resources

Chu, R., Analysis Qualifying Exam Solutions
Fremlin, D., Measure Theory
Hayes, B., Analysis Qual Solutions
Johnsrude, B., UCLA Analysis Qualifying Exam Solutions
Lott, A., UCLA Analysis Qualifying Exam Solutions
Rudin, W., Real and Complex Analysis
Folland, G., Real Analysis: Modern Techniques and Their Applications

Spring 2009
Fall 2009
Spring 2010
Fall 2010
Spring 2011
Fall 2011
Spring 2012
Fall 2012
Spring 2013
Fall 2013
Spring 2014
Fall 2014
Spring 2015
Fall 2015
Spring 2016
Fall 2016
Spring 2017
Fall 2017
Spring 2018
Fall 2018
Spring 2019
Fall 2019
Spring 2020
Fall 2020
Spring 2021

Spring 2009

Problem 1
Fubini's theorem
Problem 2
construction, Hilbert spaces, Zorn's lemma
Problem 3
Banach spaces, Hahn-Banach
Problem 4
Fatou's lemma, monotone functions
Problem 5
Lebesgue differentiation theorem
Problem 6
Fourier analysis
Problem 7
Cauchy's integral formula, Fubini's theorem
Problem 8
Picard's little theorem
Problem 9
Blaschke products, decomposition of scales, Jensen's formula
Problem 10
mean value property, Morera's theorem
Problem 11
Jordan curve theorem, winding number
Problem 12
change of variables, conformal mappings

Fall 2009

Problem 1
construction, Lp spaces
Problem 2
Fourier analysis, Laplacian, Lp spaces
Problem 3
decomposition of scales, measure theory
Problem 4
Lebesgue differentiation theorem, measure theory, regularity of Borel measures
Problem 5
construction, measure theory
Problem 6
Banach-Alaoglu, Poisson kernel, Riesz representation, weak convergence
Problem 7
Hilbert spaces, operator theory, spectral theory
Problem 8
construction, holomorphic branches
Problem 9
meromorphic functions, periodic functions, residue theorem
Problem 10
conformal mappings, construction
Problem 11
Blaschke products, Schwarz lemma
Problem 12
Jensen's formula, meromorphic functions

Spring 2010

Problem 1
construction, Lp spaces
Problem 2
harmonic functions
Problem 3
Hardy-Littlewood maximal inequality
Problem 4
Jensen's formula, subharmonic functions
Problem 5
density argument, measure theory
Problem 6
Cauchy's integral formula, Fubini's theorem
Problem 7
convex sets, Hilbert spaces
Problem 8
argument principle
Problem 9
Hilbert spaces
Problem 10
Harnack's inequality
Problem 11
Young's inequality
Problem 12
Montel's theorem, normal families, Urysohn subsequence principle
Problem 13
Banach-Alaoglu, Hahn-Banach, operator theory

Fall 2010

Problem 1
construction, dominated convergence theorem, Fatou's lemma
Problem 2
Jensen's inequality
Problem 3
density argument, Riemann-Lebesgue lemma, weak convergence
Problem 4
Hardy-Littlewood maximal inequality, operator theory
Problem 5
density argument
Problem 6
Hilbert spaces
Problem 7
Morera's theorem
Problem 8
operator theory
Problem 9
Grunsky coefficients
Problem 10
conformal mappings, open mapping theorem
Problem 11
harmonic functions, Wirtinger derivatives
Problem 12
mean value property

Spring 2011

Problem 1
Arzelà-Ascoli, Banach-Steinhaus
Problem 2
Lebesgue differentiation theorem
Problem 3
measure theory
Problem 4
Fatou's lemma, Lp spaces
Problem 5
construction, Lp spaces
Problem 6
Lebesgue-Radon-Nikodym derivative, measure theory
Problem 7
Goursat's theorem
Problem 8
subharmonic functions
Problem 9
Cantor set, Painlevé's theorem
Problem 10
Schwarz lemma
Problem 11
residue theorem, Riemann Zeta function
Problem 12
calculus

Fall 2011

Problem 1
Egorov's theorem
Problem 2
change of variables, Fourier analysis
Problem 3
convolutions
Problem 4
Cauchy-Schwarz
Problem 5
Hardy-Littlewood maximal inequality
Problem 6
Urysohn's lemma, weak convergence
Problem 7
residue theorem
Problem 8
Rouché's theorem
Problem 9
mean value property
Problem 10
Riemann mapping theorem, Schwarz lemma
Problem 11
maximum principle
Problem 12
Cauchy estimates

Spring 2012

Problem 1
construction, Lp spaces
Problem 2
measure theory
Problem 3
density argument
Problem 4
Lp spaces
Problem 5
Hilbert spaces, Riesz representation
Problem 6
Fourier analysis
Problem 7
harmonic functions, Harnack's inequality
Problem 8
construction, harmonic functions
Problem 9
Jordan's lemma
Problem 10
Gamma function
Problem 11
construction
Problem 12
Jensen's formula

Fall 2012

Problem 1
Banach-Alaoglu, Hahn-Banach, Lp spaces, Urysohn subsequence principle
Problem 2
density argument
Problem 3
Hilbert spaces, parallelogram law
Problem 4
Fourier analysis
Problem 5
Egorov's theorem, Lp spaces
Problem 6
Hardy-Littlewood maximal inequality, Lebesgue differentiation theorem
Problem 7
Schwarz lemma
Problem 8
Weierstrass factorization theorem
Problem 9
identity theorem
Problem 10
normal families, Urysohn subsequence principle
Problem 11
periodic functions
Problem 12
harmonic functions

Spring 2013

Problem 1
Lebesgue differentiation theorem
Problem 2
Hilbert spaces, measure theory
Problem 3
Lebesgue differentiation theorem
Problem 4
Banach-Alaoglu
Problem 5
harmonic functions, Hilbert spaces
Problem 6
Fourier analysis
Problem 7
subharmonic functions
Problem 8
Hadamard's three-lines theorem, operator theory
Problem 9
convex hull
Problem 10
Riemann mapping theorem
Problem 11
calculation, residue theorem
Problem 12
Banach-Alaoglu

Fall 2013

Problem 1
open mapping theorem
Problem 2
calculation
Problem 3
factor theorem
Problem 4
subharmonic functions
Problem 5
normal families
Problem 6
Cauchy estimates
Problem 7
density argument
Problem 8
AM-GM, calculation
Problem 9
Banach-Steinhaus, operator theory
Problem 10
Fourier analysis
Problem 11
Banach spaces
Problem 12
bounded variation

Spring 2014

Problem 1
Fubini-Tonelli, measure theory
Problem 2
Lp spaces
Problem 3
measure theory
Problem 4
Lp spaces
Problem 5
Banach spaces
Problem 6
Hilbert spaces
Problem 7
entire functions
Problem 8
Weierstrass factorization theorem
Problem 9
Hurwitz' theorem, Rouché's theorem
Problem 10
analytic continuation
Problem 11
calculation, residue theorem
Problem 12
calculation

Fall 2014

Problem 1
Lp spaces
Problem 2
construction
Problem 3
Lp spaces
Problem 4
Fourier analysis
Problem 5
Lebesgue-Radon-Nikodym derivative, measure theory
Problem 6
Banach spaces
Problem 7
conformal mappings
Problem 8
entire functions
Problem 9
Rouché's theorem
Problem 10
harmonic functions
Problem 11
Harnack's inequality
Problem 12
subharmonic functions

Spring 2015

Problem 1
density argument, Lp spaces
Problem 2
Lp spaces
Problem 3
Hardy-Littlewood maximal inequality
Problem 4
Fourier analysis, Hahn-Banach
Problem 5
Fourier analysis
Problem 6
Banach spaces
Problem 7
normal families, Urysohn subsequence principle
Problem 8
Phragmén-Lindelöf
Problem 9
conformal mappings, harmonic functions
Problem 10
calculation, residue theorem
Problem 11
harmonic functions
Problem 12
entire functions

Fall 2015

Problem 1
Lp spaces
Problem 2
Lp spaces
Problem 3
Lp spaces
Problem 4
Hilbert spaces
Problem 5
Baire category theorem
Problem 6
Fourier analysis
Problem 7
Schwarz reflection principle
Problem 8
periodic functions
Problem 9
normal families
Problem 10
calculation, residue theorem
Problem 11
Phragmén-Lindelöf, subharmonic functions
Problem 12
conformal mappings, harmonic functions

Spring 2016

Problem 1
density argument, Lp spaces
Problem 2
Lp spaces
Problem 3
Lebesgue differentiation theorem
Problem 4
Banach spaces, Hahn-Banach
Problem 5
Fourier analysis
Problem 6
Hilbert spaces
Problem 7
calculation, residue theorem
Problem 8
conformal mappings, normal families
Problem 9
entire functions
Problem 10
maximum principle
Problem 11
Rouché's theorem
Problem 12
harmonic functions

Fall 2016

Problem 1
Lp spaces
Problem 2
Lebesgue differentiation theorem
Problem 3
weak convergence
Problem 4
Baire category theorem, Lp spaces
Problem 5
measure theory
Problem 6
Banach spaces
Problem 7
Hilbert spaces
Problem 8
Jensen's formula
Problem 9
measure theory
Problem 10
entire functions, maximum principle
Problem 11
entire functions
Problem 12
Runge's theorem

Spring 2017

Problem 1
Lp spaces
Problem 2
Lp spaces, monotone functions
Problem 3
Banach spaces, topology
Problem 4
construction
Problem 5
density argument, Lebesgue-Radon-Nikodym derivative
Problem 6
measure theory
Problem 7
conformal mappings, Schwarz lemma
Problem 8
argument principle, Blaschke products
Problem 9
Jensen's formula
Problem 10
harmonic functions, residue theorem
Problem 11
harmonic functions
Problem 12
Cauchy estimates

Fall 2017

Problem 1
measure theory, monotone functions
Problem 2
Lp spaces
Problem 3
density argument, measure theory
Problem 4
construction
Problem 5
measure theory
Problem 6
Lp spaces, Minkowski's inequality
Problem 7
construction
Problem 8
Dirichlet problem, Poisson kernel
Problem 9
entire functions
Problem 10
Harnack's inequality, subharmonic functions
Problem 11
inverse function theorem, Schwarz lemma
Problem 12
entire functions, inverse function theorem

Spring 2018

Problem 1
Lebesgue differentiation theorem
Problem 2
Fourier analysis, Lp spaces
Problem 3
Lp spaces
Problem 4
Lebesgue differentiation theorem, Lp spaces
Problem 5
Stone-Weierstrass
Problem 6
measure theory
Problem 7
Cauchy's integral formula
Problem 8
harmonic functions
Problem 9
analytic continuation
Problem 10
meromorphic functions
Problem 11
conformal mappings, Schwarz reflection principle
Problem 12
calculation

Fall 2018

Problem 1
construction, Lp spaces
Problem 2
Stone-Weierstrass
Problem 3
weak convergence
Problem 4
measure theory
Problem 5
Baire category theorem
Problem 6
Fourier analysis
Problem 7
harmonic functions
Problem 8
Hilbert spaces
Problem 9
calculation
Problem 10
calculation, residue theorem
Problem 11
Riemann mapping theorem
Problem 12
Rouché's theorem

Spring 2019

Problem 1
calculus
Problem 2
measure theory, weak convergence
Problem 3
Baire category theorem, Lp spaces
Problem 4
Hahn-Banach, operator theory
Problem 5
Lp spaces, operator theory
Problem 6
Hilbert spaces
Problem 7
entire functions, Jensen's formula
Problem 8
Hadamard factorization
Problem 9
Blaschke products
Problem 10
calculation, residue theorem
Problem 11
harmonic functions, Schwarz reflection principle
Problem 12
Hilbert spaces

Fall 2019

Problem 1
Lebesgue-Radon-Nikodym derivative, measure theory
Problem 2
Fourier analysis
Problem 3
Lp spaces
Problem 4
measure theory
Problem 5
Hilbert spaces
Problem 6
Hahn-Banach
Problem 7
measure theory, Stone-Weierstrass
Problem 8
entire functions
Problem 9
Rouché's theorem
Problem 10
calculation, residue theorem
Problem 11
conformal mappings
Problem 12
analytic continuation

Spring 2020

Problem 1
Stone-Weierstrass
Problem 2
Helly's selection theorem, Lp spaces
Problem 3
measure theory
Problem 4
weak convergence
Problem 5
Hilbert spaces
Problem 6
Lp spaces, weak convergence
Problem 7
normal families
Problem 8
Hadamard factorization
Problem 9
inhomogeneous Cauchy integral formula
Problem 10
calculation, residue theorem
Problem 11
Runge's theorem
Problem 12
subharmonic functions

Fall 2020

Problem 1
measure theory
Problem 2
Lp spaces, weak convergence
Problem 3
construction, weak convergence
Problem 4
Banach spaces, Hahn-Banach
Problem 5
construction, Lp spaces, measure theory
Problem 6
calculus, Lp spaces
Problem 7
harmonic functions
Problem 8
Blaschke products
Problem 9
conformal mappings
Problem 10
argument principle
Problem 11
normal families, Urysohn subsequence principle
Problem 12
Cauchy's integral formula

Spring 2021

Problem 1
measure theory, Radon-Nikodym derivative
Problem 2
measure theory
Problem 3
density argument, Lp spaces
Problem 4
Arzelà-Ascoli
Problem 5
uniform boundedness principle
Problem 6
operator theory
Problem 7
harmonic functions
Problem 8
construction, entire functions, Jacobi theta function
Problem 9
Schwarz lemma
Problem 10
analytic continuation
Problem 11
entire functions
Problem 12
Jensen's formula