Math 471 Real Analysis I, Fall 2025



Instructor: Alex Iosevich
Meeting Time: MW 10:25-11:40 a.m.
Textbook: Measure, Integration and Real Analysis, by S. Axler (link to Axler's website)
Office Hours: MW 9:00-10:00 a.m.
Grading: Homework (%30), Midterm (%30), and Final (%40)

Homework policy: Homework will be assigned at the end of every class, due before the start of the following class. Only one or two simple problems will be assigned from Monday to Wednesday, with the bulk of the assignment due on Monday morning. No late homework will be accepted, but the lowest %20 of homework scores will be dropped.

Midterm: The midterm will be in the evening, on a day that works with everybody's schedule. This way I can give you three hours to complete the exam to take time pressure out of the process.

Homework #1 (due Wednesday, August 27 on Gradescope):

Problem 1: page 8, problem 8
Problem 2: page 8, problem 12
Problem 3: Prove that eiθ=cos(θ)+isin(θ)e^{i \theta}=\cos(\theta)+i\sin(\theta)  . Do this by showing that both sides satisfies the differential equation y″+y=0,y′(0)=i,y(0)=1. Prove that the solution to this initial value problem is unique.
y''+y=0, y'(0)=i, y(0)=1.

Homework # 2 (due Wednesday, September 3 on Gradescope)

Problem 4: page 12, problem 1
Problem 5: page 12, problem 2
Problem 6: page 12, problem 4
Problem 7: page 12, problem 5
Problem 8: page 23, problem 1
Problem 9, page 23, problem 2
Problem 10: page 23, problem 4
Problem 11: page 23, problem 9
Problem 12: page 24, problem 10
Problem 13: page 24, problem 13
Problem 14: Let f(x)=sin(x)x.f(x)=\frac{\sin(x)}{x}. Prove that this function is Riemann integrable on any interval [1,R],1<R<∞[1,R], 1<R<\infty. Furthermore, prove that
|∫1Rf(x)dx|≤10,\left| \int_1^R f(x) dx \right| \leq 10, for any R>1.
R>1.

Problem 15: Compute ∑k=1∞k2k by using the technique we learned in class. More precisely, write k=∑i=1k1,k=\sum_{i=1}^k 1, reverse the order of summation, and go from there.

Homework # 3 (due Monday, September 8 on Gradescope)

Problem 16: page 38, problem 1
Problem 17: page 38, problem 3
Problem 18: page 38, problem 8
Problem 19: page 38, problem 10
Problem 20: page 38, problem 12
Problem 21: page 39, problem 17
Problem 22: page 39, problem 18
Problem 23: Compute ∑k=1∞k22k\sum_{k=1}^{\infty} \frac{k^2}{2^k}    . Please use the summation method we used to compute ∑k=1nk2.\sum_{k=1}^n k^2.

Homeowrk # 4 (due Wednesday, September 10 on Gradescope)

Problem 24: page 39, problem 22
Problem 25: page 40, problem 30


Homework # 5 (due Monday, September 15 on Gradescope)

Problem 26: page 45, problem 1
Problem 27: page 44, problem 2
Problem 28: page 45, problem 5
Problem 29: page 45, problem 9
Problem 30: page 45, problem 10
Problem 31: page 60, problem 1
Problem 32: Find all positive real numbers a such that the sum ∑n∊A1na\sum_{n \in A} \frac{1}{n^a}  converges, where A is the set of positive integers that do not have 3 and 5 in their base 10 expansions.


Homework # 6 (due Wednesday, September 17 on Gradescope)

Problem 33: page 61, problem 14
Problem 34: page 61, problem 15


Homework # 7 (due Monday, September 22 on Gradescope)

Problem 35: page 61, problem 11
Problem 36: page 61, problem 19
Problem 37: page 61, problem 22 a)
Problem 38: page 61, problem 22 b)
Problem 39: page 71, problem 1
Problem 40: page 71, problem 2
Problem 41: page 71, problem 3
Problem 42: page 71, problem 5
Problem 43: page 71, problem 8


Homework # 8 (due Monday, September 29 on Gradescope)

Problem 44: page 72, problem 12
Problem 45: page 72, problem 14
Problem 46: page 72, problem 15
Problem 47: page 84, problem 1
Problem 48: page 84, problem 2
Problem 49: page 84, problem 3
Problem 50: page 85, problem 5
Problem 51: page 85, problem 7
Problem 52: page 85, problem 8
Problem 53: page 85, problem 9
Problem 54: page 86, problem 13


Homework # 9 (due Wednesday, October 1 on Gradescope)

Problem 55: page 87, problem 18
Problem 56: page 87, problem 19


Homework # 10 (due Monday, October 6, 2025)

Problem 57, page 99, problem 1
Problem 58, page 99, problem 2
Problem 59, page 99, problem 5
Problem 60, page 100, problem 12
Problem 61, page 100, problem 14
Problem 62 Prove that if f is a non-negative twice differentiable function such that f″(x)≥0,f''(x) \ge 0, then for x<y,0≤t≤1,x<y, 0 \leq t \leq 1, we have f((1−t)x+ty)≤(1−t)f(x)+tf(y). Please give a direct proof and do not appear to any general theorems.
f((1-t)x+ty) \leq (1-t)f(x)+tf(y). x<y

Problem 63 Let A be an n by n matrix of 1s and 0s such that if two 1s are present in a given row, they cannot be present in the same slots in any other row. Get the best upper bound you can on the number of 1s in A.

Homework # 11 (due Wednesday, October 8, 2025 on Gradescope)

Problem 64, page 106, problem 1
Problem 65, page 106, problem 2
Problem 66 Let f:ℤ→ℝf:{\mathbb Z} \to {\mathbb R}   . Suppose that ∑n∊ℤ|f(n)|<∞.\sum_{n \in {\mathbb Z}} |f(n)|<\infty. Define the maximal function of f analogously to the Hardy-Littlewood maximal function, and prove the corresponding discrete version of the Hardy-Littlewood maximal inequality.


Homework # 12 (due Wednesday, October 15 on Gradescope)

Problem 67, page 106, problem 5 and problem 6
Problem 68, page 106, problem 7
Problem 69, page 106, problem 8
Problem 70, page 106, problem 9
Problem 71, page 107, problem 10
Problem 72, page 107, problem 14
Problem 73, page 116, problem 1
Problem 74, page 116, problem 3
Problem 75, page 116, problem 5
Problem 76, page 116, problem 8
Problem 77, page 116, problem 9


Homework # 13 (due Monday, October 20 on Gradescope)

Problem 78, page 128, problem 1
Problem 79, page 128, problem 3
Problem 80, page 128, problem 9
Problem 81 Let S be a subset of three dimensional Euclidean space consisting of n points. Prove that n2≤#π1(S)⋅#π2(S)⋅#π3(S),  where  π1(x)=(x2,x3),etc.n^2 \leq \# \pi_1(S) \cdot \# \pi_2(S) \cdot \# \pi_3(S), where \pi_1(x)=(x_2, x_3), etc. \pi_1(x)=(x_2, x_3),
\pi_1(x)=(x_2,x_3), and define \pi_2, \




Preparing for this class:

This is a graduate level course in measure theory. In order to get the most out of this course, I recommend reviewing undergraduate analysis, linear algebra, and, just as importantly, practicing applying these concepts in a direct and elementary way. Among the elementary concepts I recommend reviewing is basic inequalities such as the Cauchy-Schwarz inequality, Young's inequality, and Jensen's inequality.

To give you an illustration of what I mean by applying those inequalities, consider the following problem. Consider an n by n matrix AA with 1-0 entries entries {aij}1≤i,j≤n{\{a_{ij} \}}_{1 \leq i,j \leq n}   satisfying aijaij′=1a_{ij} a_{ij'}=1  for at most one value of i if j¬=j′.j \not=j'. What is the largest possible number of 1's as a function of n in such a matrix, up to a constant? Can you construct a matrix that realizes this bound? Can you construct a geometric problem where this calculation naturally comes up?

One of my main goals when I teach is to bridge the gap between classwork and research. When you work on a research problem, it typically does not come with a sign in neon lights announcing which techniques from your graduate courses you need to use in order to solve it. Furthermore, even if you identify the right technique, you typically need to perform quite a bit of technical work to execute the idea. This type of bridging will be emphasized throughout the course.

Here is another example of what I mean by reviewing elementary concepts. The following problem can be found in Edmund Landau's calculus book as an exercise. Let f∊C1(a,b),f \in C^1(a,b), and suppose that f′(x)≥1.f'(x) \ge 1. Further suppose that f′f'       is monotonic. Can you prove that |∫abeiRf(x)dx|≤4|R|?\left| \int_a^b e^{iRf(x)} dx \right| \leq \frac{4}{|R|}? (a
Can you further show that if f is twice continuously differentiable on [a,b][a,b]  , and f″(x)≥1f''(x) \ge 1   , then
|∫abeiRf(x)dx|≤C′|R|?\left| \int_a^b e^{iRf(x)} dx \right| \leq \frac{C'}{\sqrt{|R|}}?
Another great problem to review is the following. Can you determine all complex values z, with |z|=1,|z|=1, such that
∑n=1∞znn\sum_{n=1}^{\infty} \frac{z^n}{n} converges conditionally?

Another aspect of analysis I would like to emphasize, to the extent the time and curriculum requirements allow, is applications to related areas of mathematics. Here is an example of a fun problem to think about, if you have not seen it before. Consider Ω,\Omega, a bounded closed region in d-dimensional Euclidean space of area k,k, a positive integer. Can prove that in a "typical" translate Ω+τ\Omega+\tau  , where the notion of "typical" is left for you to nail down the number of  of integer lattice points is ≤k?

Here is another elementary example worth recalling. Suppose that C is a closed twice differentiable convex curve in the plane parameterized by ∫01k(t)dt=2π?\int_0^1 k(t) dt= 2\pi ? Now redefine the curvature function as follows. Let N(x) denote the function from C to the unit circle where every point x on C is taken to the outward unit normal at x. Prove that k(t)=det(dN)x, where t is the parameter corresponding to the point x on C in the parameterization of C given above. k(t)=det(dN)_x,
\leq k?