Chapter 6, 10
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Let \(X,Y\) be normed vector spaces and \(F:X\to Y\) a function.
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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.
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Show that \(F(x)=o(1)\) as \(x\to a\) if and only if \(F(x)\to 0\) as \(x\to a\).
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Show that F is continuous at a if and only if \(F(x)=F(a)+o(1)\).
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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 \(x\to a\).
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Section 6.1: Exercises 1, 2, 3, 8
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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.
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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.
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Section 6.2: Exercises 1, 2, 4, 7, 10
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Exercise 6.3.1
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Section 6.7: Exercises 1, 3, 4, 8
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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 \(F'(x)=(G'(x),H'(x))\).
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Problems 6.8.2, 5.
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Section 6.6: Problems 4, 6
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Section 6.8: Problem 12
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Section 10.1: 5, 7, 9, 10
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Section 10.2: 1, 2
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Section 10.4: 1, 2, 3, 6
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Exam problems
The following is a list of problems from earlier exams that will be a good preparation for the exam.
- MAT2400, 2019 (try not to look at the solutions)
- MAT2400, 2018, with solutions
- MAT2400, 2017: 1, 2, 4, with solutions
- MAT2400, 2016, with solutions
- MAT2400, 2015: 1, 3, 4, 5, with solutions
- MAT2400, 2014 (apart from 3 and 4d), with solutions