Exercises

These are recommended, but not mandatory, exercises. You should do as many of these as you can every week. And remember: you will get much more out of these exercises if you discuss them with your fellow students!

Problems for Chapter 1–5

Chapter 6, 10

  • Wed 08.04

    • Let \(X,Y\) be normed vector spaces and \(F:X\to Y\) a function.

      • Show that \(F(x)=O(1)\) as \(x\to a\) if and only if there is a ball centered at a where F is bounded.

      • Show that \(F(x)=o(1)\) as \(x\to a\) if and only if \(F(x)\to 0\) as \(x\to a\).

      • Show that F is continuous at a if and only if \(F(x)=F(a)+o(1)\).

      • Show that if \(F(x)=o(\alpha(x))\) and \(f(x)=O(\beta(x))\) as \(x\to a\) for some \(\alpha,\beta: X\to[0,\infty)\) and \(f:X\to \mathbb{R}\), then \(f(x)F(x)=o(\alpha(x)\beta(x))\) as \(x\to a\).

    • Section 6.1: Exercises 1, 2, 3, 8

    • Show that a function \(f:\mathbb{R}\to\mathbb{R}\) is Fréchet differentiable if and only if it is differentiable in the usual sense that you learnt in Calculus.

    • Let \(f:\mathbb{R}\to Y\) be a function, where Y is a normed vector space. Explain why we can think of the Fréchet derivative \(f'(a)\) as a vector in Y.

    • Section 6.2: Exercises 1, 2, 4, 7, 10

  • Wed 15.04

    • Exercise 6.3.1

    • Section 6.7: Exercises 1, 3, 4, 8

  • Wed 22.04

    • Let \(X,Y,Z\) be normed vector spaces and let \(F:X\to Y\times Z\) be a function. Denote its components by \(F(x)=(G(x),H(x))\). Show that F is Fréchet differentiable if and only if G and H are, and show that \(F'(x)=(G'(x),H'(x))\).

    • Problems 6.8.2, 5.

  • The rest of the semester

    • Section 6.6: Problems 4, 6

    • Section 6.8: Problem 12

    • Section 10.1: 5, 7, 9, 10

    • Section 10.2: 1, 2

    • Section 10.4: 1, 2, 3, 6

Exam problems

The following is a list of problems from earlier exams that will be a good preparation for the exam.

Published Apr. 7, 2020 1:06 PM - Last modified Aug. 19, 2021 9:20 AM