Real Analysis Qual Prep

Jul 29 - Sep 11

Meeting times: Refer to email sent.

Official Qualifying Exam Date and Time: Monday, September 22 at 1-4pm!

If you have any additional questions feel free to email me or find me in my office.

Weekly Outline:

1. Undergraduate

   • Continuity 

   • Convergence

   • Derivatives

   • Integration

2. Real A

   • Convergence Theorems

   • Measures

   • Measurable sets and functions

   • Limits

   • Fubini-Tonelli

3. Real B

4. Real C

5. Mock Qual

6. Reflection of Mock Qual

7. Mock Qual

Meeting 1 (Jul 29) - Continuity and Convergence worksheet 

Key questions are the ones that state prove and disprove.

Meeting 2 (Jul 31) - Derivatives and Integration worksheet (I would suggest solving the Riemann Integrable questions to help remind of definitions.)

Problems presented: 1, 11, 12, 16 from Continuity and Convergence

Link to wiki for Thomae's Popcorn Function, which can be used for problem 7 on continuity and convergence. If a solution is needed for a reference please email me.

Meeting 3 & 4 (Aug 5, Aug 7) - Real A worksheet 

I highly recommend completing all problems from Convergence Theorems and the first problem from Fubini-Tonelli.

Problems presented (Aug 5): 1, 5, 7, 10 from Derivatives and Integration

Problems presented (Aug 7, Real A):  3, 4 from Convergence Theorems, 1 from Limits, 1 from Fubini-Tonelli

Meeting 5 & 6 (Aug 12, Aug 14) - Real B Worksheet 

I recommend doing the second problem for LRN, as this problem personally helped me truly understand the theorem. 

Problems presented (Aug 12) - 2,3 from Measurable Sets/Functions, 2 from Measures

Problems presented (Aug 14) - 1 from Banach Space, 1, 2 from LRN, 1 from Lebesgue Differentiation, 8 from Misc., 

Meeting 7 & 8 (Aug 19, Aug 21) - Real C Worksheet 

Please do all the "recommended problems".

Problems presented (Aug 19) - 1 from BVAC, 4 from Topology Theorems, 2 from LD

Hints for problems from the recommended problem list:

1. One direction should be clear, the other you need to define a linear functional on continuous functions, then apply Hahn Banach to extend to the entire space. After, show that it cannot be represented by integration.

2.  Extend f to a periodic sequence, then compute fourier coefficients explicitly. Utilize plancherel and parseval identity.

3. Similar proof to show one can write a Hilbert Space as a closed set direct sum the closed set perp.

4. Refer to email.

5. I think this problem should be clear.

6. This problem should be clear too.

7. Reduce the problem so you may assume positive and sup norm equal to 1. After, relate L^p, L^q norm of f to bound the limsup from above. Using the L^q norm of f directly lower bound using a "nice" set. This will bound the liminf from below.

8. Recall that Schwartz functions are in L^p for p in [1,\infty].

9. Trivial with Fourier inverse.

10. Reduce problem to assume f hat has a neighborhood locally equal to 0 around 0. Directly compute the Fourier coefficients and use taylor series of e.

Homework (Aug 25 - 28) - Please take the mock qual on your own time at some point between the dates listed. I will proctor a mock qual session on Aug 27 from 1:30 - 4:30pm in Skye 281. Check your email for more information.

Meeting 9 (Aug 26) - Please bring any questions and we will work as a group to figure out how to solve the problems.

Meeting 10 (Aug 28) - Reflection on Mock Qual and solutions.

Meeting 11 & 12 (Sep 2, Sep 4) - Extra problems on Schwartz space and distributions.

This week we will work on any problems that need reviewing.

Meeting 13 (Sep 9) - Mock Qual 2  (will be uploaded Sunday morning on Sep 7 and will be sent via email)

Meeting 14 (Sep 11) - Reflection on Mock Qual 2 and solutions.