TMA4225 Analysens grunnlag – Foundations of Analysis

We post the exercises in inverse chronologial order (newest on top).

• Ch 12: 10,15

• Ch 5C: 13
• Ch 7A: 4,5
• Ch 12: 3,4

• Ch 7A: 1, 2, 7, 8
• Ch 7B: 8

• Ch 5B: 1
• Ch 5C: 8, 12, 14

• Ch 4B; 3, 5, 6
• Ch 5A: 2, 9, 10

• Ch 3B; 10, 11, 16(a)
• Ch 4A: 1, 2, 9, 10
• Hint for 3B–10: Consider \(f^{-1}([1,\infty))\) and \(f^{-1}([0,1))\) separately
• Hint for 16(a): Approximate by a continuous function – and be careful with your estimates

• Ch 3A: 4, 6, 9, 17
• Ch 3B: 3, 5, 12
• Suggestion for 3A–4: Put \(f(x)=1\) for all irrational \(x\), then do somethiing clever on the rational numbers
• Hint for 3A–17: Put \(\hat f_k(x)=\inf\{f_j(x): j \ge k\}\) and note that \(\hat f_1, \hat f_2, …\) is an increasing sequence of functions converging pointwise to \(f\).

In case you don't know the inferior limit of a sequence of numbers, the definition is \[ \liminf_{k \to \infty} a_k = \lim_{k \to \infty} \inf\, \{ a_j : j \ge k\}. \] The limit on the right exists because it is a limit of an increasing sequence.

Problem 3A–17 is known as Fatou's lemma. It is an indispensible tool in analysis, even if not heavily used in the present text.

Axler:
Ch 2D: 5, 6 (this is Littlewood's first principle: Every measurable set is almost a finite union of intervals)
Ch 2E: 5, 12
Ch 3A: 1, 2, 3

Axler:
Ch 2B: 18, 19, 24, 27
Ch 2D: 19

Axler:
Ch 2B: 6, 7, 11
Ch 2C: 1, 5
Ch 2D: 5, 12

Axler:
Chap 1A: 4, 13.
Chap 1B: 1.
Chap 2A: 1, 2, 4, 7, 9, 11.
Chap 2B: 1, 2, 11.
Extra problem: Where is page 22?

Here is a solution to 2A problem 11. This will be useful later!

Try to do as many of these as possible. In the exercise session we will concentrate on the numbers in bold.
Supplement:
Section C: 2, 5, 6, 10, 13,14.
Section D: 8, 9, 13, 15, 16.
Section E: 6.
Axler:
Chap 1A: 1, 4, 13.
Chap 1B: 1, 5.
Chap 2A: 1, 2.