MAT2400 - Real Analysis

Exercises

Below is a regularly updated list of exercises for the course, with each date indicating when the exercises will be reviewed in the plenary session. The plan is to update the list late on Fridays. For a broader view of the course's planned exercises, see the tentative list [More].

Old solutions to exercises can be found here [More].

Jan 26

1.1. Proofs: Exercises 1�C4.

1.2. Sets and Boolean operations: Exercises 1�C9.

1.3. Families of sets: Exercises 1��7.

Answers [More].

Feb 2

1.4. Functions: Exercises 1, 2, 4, 6, 7, 8, 9.

1.6. Countability: Exercises 1, 2, 4, 5.

2.1. Epsilon-delta and all that: Exercises 1, 2, 4, 6.

There is a typo in 1.4.7a, B?Y should read B?f(X).

Answers [More].

Feb 9

2.2. Completeness: Exercises 1, 2, 4, 5.

2.3. Four important theorems: Exercises 1, 2, 3, 4, 9.

3.1. De?nitions and examples: Exercises 5, 6, 7, 9, 12.

3.2. Convergence and continuity: Exercises 1, 4, 5, 6, 8.

Answers [More].

Feb 16

3.3. Open and closed sets: Exercises 1, 4, 5, 6, 7, 10, 11, 12.

3.4. Complete spaces: Exercises 1, 2, 3, 5, 6, 7, 9.

Answers [More].

Feb 23

3.5. Compact sets: Exercises 1, 2, 3, 4, 5, 7, 9, 12, 13, 16.

3.6. An alternative description of compactness: Exercises 1, 3, 5, 7.

There is a misprint in the hint in problem 7: ��? should be =?.

Answers [More].

Mar 1

4.1. Modes of continuity: Exercises 1, 2, 3, 4.

4.2. Modes of convergence: Exercises 1, 3, 4, 6, 9, 10.

4.3. Integrating and differentiating sequences: Exercises 1, 2, 3, 6, 8, 10.

In Exercise 4.3.6, there is an error in the text. The statement ?converges uniformly on R? should be corrected to ?converges uniformly on [a, b], for any a<b?.

Answers [More].

Mar 8

4.4. Applications to power series: Exercises 1, 2, 3, 4, 5, 6.

4.5. Spaces of bounded functions: Exercises 1, 2, 3, 4, 5, 6.

4.6. Spaces of bounded, continuous functions: Exercises 1, 2, 3.

Answers [More].

Mar 15

4.7. Applications to differential equations: Exercises 1, 2, 3.

4.8. Compact sets of continuous functions: Exercises 1, 2, 3 , 4, 5, 7, 8.

Answers [More].

Apr 5

4.9. Differential equations revisited: Exercises 1, 2.

4.10. Polynomials are dense in the continuous function: Exercises 1, 2, 3 , 4, 5.

Answers [More].

Apr 12

5.1. Normed spaces: Exercises 4, 5, 6, 8, 9, 10, 11, 12.

5.2. In?nite sums and bases: Exercises 1, 2, 3.

Answers: [More].

Apr 19

5.3. Inner product spaces: Exercises 1, 2, 3, 4, 5, 6, 8, 9, 11, 13.

5.4. Linear operators: Exercises 1, 2, 3, 4, 5, 6, 7, 8.

Answers [More].

Apr 26

5.5. Inverse operators and Neumann series: Exercises 3, 4, 7.

Exam23-4.

6.1. The derivative: Exercises 3, 5, 11.

6.2. Finding derivatives: Exercises 1, 4, 5, 6, 7, 9, 10.

Exam21-7a, Exam19-1, and Exam18-1.

If little time, omit Exercises 6.2.9 and 6.2.10, substituting them with the three problems from earlier exams.

Answers [More].

May 3

6.3. The Mean Value Theorem: Exercises 1, 2.

6.4. The Riemann Integral: Exercises 1, 2, 3.

6.5. Taylor��s Formula: Exercises 2, 3, 4, 5, 6.

6.7. The Inverse Function Theorem: Exercises 1, 2, 3.

Answers [More].

May 10

Exam 2022, Exam 2023.

If time permits,

10.1. Fourier coefficients and Fourier series: Exercises 1, 3, 5, 6, 7, 9, 10, 11.

10.2. Convergence in mean square: Exercises 1, 2.

Answers (10.1, 10.2) [More].

May 24

Exam 2021.

Published Dec. 2, 2023 7:47 AM - Last modified May 11, 2024 6:54 AM