Solutions
Tom Lindstr?m wrote solutions to many of the problems when he taught the course in 2016; these are available here. Note that at that time, the book was still a set of lecture notes (see here), so some of the problems might have different numbering.
There are also solutions (in norwegian) from 2015 here. The exercises are mostly the same as in Spaces, but with a different numbering found here.
Chapter 1
- Wed. 15.01
- Section 1.1: Exercise 1, 2, 3. (In 1.1.1, prove the statement both by reductio ad absurdum and by proving the contrapositive.)
- Section 1.2:
- Exercises 1-4
- Prove Propositions 1.2.1 and 1.2.2. (Have a peek at the proof in the book only if you need a hint.)
- Exercise 8
- Section 1.3: Exercise 1-4 and 7.
- Thurs 16.01
- Section 1.4: Exercises 1-8
- Section 1.6: Exercises 1-4. (Hint for 1.6.4: Show first the following claim: There exist bijective functions \(g : \mathbb{N}\to A\) and \(h : \mathbb{N}\to B\).)
Chapter 2
- Wed 22.01
- ????(Before the lecture) Let \(p>0\) be a given number and define the sequence \(x_n = n^{-p}\). Show that \(x_n \to 0\) as \(n \to \infty\).
- ????(Before the lecture) Show that the functions \(x \mapsto |x|\) and \(x \mapsto x^2\) (for \(x\in\mathbb{R}\)) are continuous.
- Section 2.1: Exercises 1, 2, 4, 7
- Section 2.2: Exercises 1-5, 10, 9
- Section 2.3: Exercises 1, 2, 4, 9
- Prove that a sequence of vectors converges if and only if each component converges. In other words: Let \(\{x_n\}_{n\in\mathbb{N}}\) be a sequence in \(\mathbb{R}^m\) and let \(x\in\mathbb{R}\) be given. Write the components of \(x_n\) as \(x_n=(x_n^{(1)}, \ldots, x_n^{(m)})\). Prove that \(x_n\to x\) as \(n\to\infty\) if and only if \(x_n^{(i)} \to x^{(i)}\) as \(n\to\infty\) for every \(i\in\{1,\ldots,m\}\).
Hint: Use the fact that there exists constants \(C_1,C_2>0\) such that \(\|x\|\leq C_1\sum_{i=1}^m |x^{(i)}|\) and \(\sum_{i=1}^m |x^{(i)}| \leq C_2 \|x\|\), for every \(x\in\mathbb{R}^m\). (This is a fact from linear algebra.) - (If you have the time) Show that any convergent sequence in \(\mathbb{R}^m\) is also Cauchy.
Chapter 3
- Thurs 23.01
- ????(Before the lecture) Prove Proposition 3.1.4 (the inverse triangle inequality): If \((X,d)\) is a metric space and \(x,y,z\in X\) then \(|d(x,y)-d(y,z)|\leq d(x,z)\).
- Section 3.1: Exercises 1, 2, 6, 7
- Section 3.2: Exercises 1, 2, 5, 6, 8
- Wed 29.01
- Consider \(X=\mathbb{R}\) with the canonical metric \(d(x,y)=|x-y|\). Show that the interval \((a,b)\) is open and that \([a,b]\) is closed, for any \(a,b\in\mathbb{R}\) with \(a\leq b\). (What happens if \(a=b\)?) Explain why \((a,b]\) is neither open nor closed when \(a<b\).
- Consider the metric space \(X = \mathbb{R}^n\) with the Euclidean metric. Let \(x\in X\) and \(r>0\), and define \(B=B(x;r)\). Determine \(B^\circ\), \(\overline{B}\) and \(\partial B\), the interior, closure and boundary of \(B\), that is, find an expression of the form \(B^\circ = \{y\in X\ |\ \dots\}\) for the three sets \(B^\circ\), \(\overline{B}\) and \(\partial B\).
- Let \((X,d)\) be a metric space. Prove that every finite subset of \(X\) is closed. (A finite set is a set containing only finitely many points.)
Hint: Show first that singletons (i.e. sets containing only one point) are closed. Next, use Proposition 3.3.13 b). - Section 3.3: Exercise 1, 2, 3, 11, 7
- Thurs 30.01
- Section 3.3: Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces, let \(f:X\to Y\) be a continuous function, let \(y\in Y\) be some given point, and consider the problem of finding an \(x\in X\) such that
\(f(x)=y\)
(that is, we wish to solve the above equation). Prove that the set of solutions of this equation is a closed subset of \(X\).
Hint: Phrase the question in terms of finding \(f^{-1}\) of a closed set. - Consider \(X=\mathbb{R}\) with the canonical metric \(d(x,y)=|x-y|\). The support of a function \(f:\mathbb{R}\to\mathbb{R}\) is the set of points \(x\) where \(f(x)\neq 0 \). Prove that the support of a continuous function is always open.
Note: In the literature, the "support" of \(f\) is usually defined to be the closure of the above set, which is of course closed, not open. - Section 3.4: Prove that \((\mathbb{Z},d)\), where \(d(x,y)=|x-y|\), is complete.
Hint: What does it mean for a sequence in \((\mathbb{Z},d)\) to converge or to be Cauchy? - Section 3.4: Exercise 2, 4, 6
- Section 3.3: Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces, let \(f:X\to Y\) be a continuous function, let \(y\in Y\) be some given point, and consider the problem of finding an \(x\in X\) such that
- Wed 05.02
- (Before the lecture) Let \((X,d)\) be a metric space where \(X\) is finite (that is, \(X\) contains only finitely many points). Prove that \(X\) is compact – that is, every sequence in \(X\) has a convergent subsequence.
- Prove Proposition 3.5.1.
- Section 3.5: Exercises 5, 6, 7, 8 (In exercise 5(a), the subsequences mentioned in the text might converge to a point outside of \(A\).)
- Thurs 06.02
- Section 3.5: Exercises 9, 11, 12, 15
Chapter 4
- Mon 10.02
- Section 4.1: Exercise 2, 4
- A function \(f:\mathbb{R}\to\mathbb{R}\) is H?lder continuous if there is some \(\alpha\in(0,1]\) (called the H?lder exponent) and some \(C>0\) such that \(|f(x)-f(y)|\leq C|x-y|^\alpha\) for all \(x,y\in \mathbb{R}\). Prove that every H?lder continuous function is uniformly continuous.
- If you are wondering why we restrict to \(\alpha\leq 1\): Prove that if \(|f(x)-f(y)|\leq C|x-y|^\alpha\) for all \(x,y\) for some \(\alpha>1\), then \(f\) is constant. Try first with \(x=0,y=1\) and estimate \(|f(0)-f(1)|\leq \sum_{i=1}^n |f(i/n)-f((i-1)/n)|\). Then let \(n\to\infty\).
- Section 4.2: Prove that if \(\{f_n\}_n\) is a sequence of uniformly continuous functions converging uniformly to some function \(f\), then \(f\) is also uniformly continuous.
- Section 4.2: Exercises 1, 2
- Show that the function \(f:\mathbb{R}\to\mathbb{R},\ f(x)=|x|\), is Lipschitz continuous.
- Wed 12.02
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Section 4.2: Exercises 7, 9
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Section 4.5: Exercises 1, 2, 7
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Section 4.6: Exercises 2, 3
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The space \(C_b(X,Y)\) is always "larger" than \(Y\), in the sense that \(Y\) can be embedded in \(C_b(X,Y)\):
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Indeed, show that the map \(i:Y\to C_b(X,Y)\) which maps \(y\in Y\) to the constant function \(f(x)\equiv y\), is an embedding (cf. Definition 3.1.3).
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Show that \(i(Y)\) is precisely the subset of constant functions, and that this set is a closed subset of \(C_b(X,Y)\).
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Conclude that \(C_b(X,Y)\) is complete if and only if \(Y\) is complete.
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Section 4.7: Let f be the function \(f(y,t)=y\). Choose initial data, say, \(y_0=1\). Perform a fixed point iteration of the equation
\(y(t) = y_0 + \int_0^t f(y(s),s)\,ds\)
that is, for some continuous function \(y^0\) (here, the superscripts are indices, not powers) let
\(y^{n+1}(t) = y_0 + \int_0^t f(y^n(s),s)\,ds\)
for \(n=0,1,2,\dots\). (It's easiest to start with \(y^0\equiv0\).) Compute \(y^1,y^2,y^3,y^4\). Give an expression for \(y^n\) for any \(n\), and prove that \(y^n\to y\), where \(y(t)=e^t\).
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Thurs 20.02
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Prove that \(\int_0^x \cos(y) dy = \sin(x)\). To do so, start with the definitions
\(\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n},\qquad \sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}. \)
Prove that the Taylor series for \(\cos\) converges uniformly in any interval \([0,a]\) (use Weierstrass' M-test), and then apply Corollary 4.3.3. -
Let \(\{v_n\}_n\) be a sequence of functions \(v_n\in C^2([a,b],\mathbb{R})\) (all twice continuously differentiable functions) and assume that:
(i) \(\sum_{n=1}^\infty v_n''\) converges to some \(h\in C([a,b],\mathbb{R})\)
(ii) there is some \(x_0\in[a,b]\) such that \(\sum_{n=1}^\infty v_n(x_0)\) converges
(iii) there is some \(x_1\in[a,b]\) such that \(\sum_{n=1}^\infty v_n'(x_1)\) converges.
Prove that \(f(x)=\sum_{n=1}^\infty v_n(x)\) converges, and that \(f'=\sum_{n=1}^\infty v_n'\) and \(f''=\sum_{n=1}^\infty v_n''\). -
Give counterexamples to the previous problem when we do not assume either (ii) or (iii).
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Wed 26.02
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Problem 1–4 in these notes.
Hint for problem 4: Use the expression \(\sin(y)=\frac{e^{iy}-e^{-iy}}{2i}\) and write \(f\) in terms of exponential functions. You might not end up with a very nice expression. -
Exercise 4.4.1
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Thurs 27.02
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Exercise 4.8.6, 7
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Thurs 05.03
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Section 4.10: 1, 2, 3
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Given \(f\in C([0,1],\mathbb{R})\), the n-th order Bernstein approximation of f is the polynomial
\(f_n(x) = \sum_{k=0}^n {n \choose k} x^k (1-x)^{n-k} f(k/n).\)
Using Matlab, Python or similar, compute and plot the Bernstein approximation of the following functions for various choices of n: \(f(x)=1,\ f(x)=x,\ f(x)=x^2,\ f(x)=|x-1/2|,\ f(x)=\sqrt{|x-1/2|}.\)
Prove that for the first two functions, \(f=f_n\).
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Chapter 5
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Wed 11.03
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Prove the following fact (Proposition 5.1.3): If \((V,\|\cdot\|)\) is a vector space then the function \(d\) defined by \(d(x,y)=\|x-y\|\) is a metric on V. (The metric d is the metric induced by the norm \(\|\cdot\|\), and \((V,d)\) is the metric space induced by the vector space \((V,\|\cdot\|)\).)
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Let V be a nontrivial vector space (meaning that \(V\neq \{0\}\)). Prove that the discrete metric on V is not induced by any norm on V. Find another example of a metric which is not induced by any norm.
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Prove Proposition 5.1.4.
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Exercise 5.1.10, 5.1.11.
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Wed 25.03
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Let \(V = \ell_\infty(\mathbb{R})\), equipped with the usual norm \(\|u\|_{\ell_\infty} = \sup_{i\in\mathbb{N}}|u_i|\). Let \(e_i = (0,\dots,0,1,0,\dots)\), that is, the element in \(\ell_\infty(\mathbb{R})\) with 1 in the i-th component and all other components equal to 0. In the lecture we proved that \(\{e_i\}_{i\in\mathbb{N}}\) is not a basis for V, from the fact that \(s_n\not\to u\) as \(n\to\infty\), where \(u=(1,1,\dots)\) and \(s_n=\sum_{i=1}^n u_i e_i\). Find a necessary and sufficient condition on a general element \(u\in V\) that ensures that \(s_n\to u\) as \(n\to\infty\).
Hint: Write down an expression for \(\|u-s_n\|_{\ell_\infty}\) first. -
Let \(\mathcal{P}([0,1]) = \Big\{p:[0,1]\to\mathbb{R}\ |\ \exists\ a_0,\dots,a_k \in \mathbb{R} \text{ s.t. } p(x)=\sum_{i=0}^k a_ix^i\Big\}\), the set of all polynomials on the unit interval [0,1]. Equip \(\mathcal{P}([0,1])\) with the supremum norm \(\|p\|_\infty = \sup_{x\in[0,1]}|p(x)|\).
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Explain why \(\mathcal{P}([0,1])\) is infinite-dimensional (that is, explain why it is not finite-dimensional).
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Show that \(\{e_i\}_{i=0,1,2,\dots}\) is a basis for \(\mathcal{P}([0,1])\), where \(e_i\) is the function \(e_i(x) = x^i\).
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Section 5.3: Exercises 1, 2
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Thurs 26.03
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Section 5.3: Exercises 8, 9, 10, 12
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Section 5.4: Exercises 2, 3
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Wed 01.04
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Section 5.4: Exercises 4, 5, 7
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(This is a generalization of exercise 5.4.8.) We aim to show that if V and W are normed vector spaces and V is finite-dimensional, then every linear operator \(A:V\to W\) is bounded. Thus, only linear operators defined on an infinite-dimensional space can be unbounded. Proceed as follows:
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Let \(e_1,\dots,e_n\) be a basis for V. Recall from the lecture on 12 March that \(\|\cdot\|_V\) is equivalent to the norm \(|||u||| = \sum_{i=1}^n |\alpha_i|\), where \(\alpha_1,\dots,\alpha_n\) are the unique scalars such that \(u=\sum_{i=1}^n \alpha_i e_i\). Prove that \(\|A(u)\|_W \leq d|||u|||\) for every \(u\in V\), where \(d=\max_{i=1,\dots,n} \|A(e_i)\|_V\).
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Conclude that there is some \(C>0\) such that \(\|A(u)\|_W \leq C\|u\|_V\) for every \(u\in V\).
Hint: Use the fact that \(|||\cdot|||\) and \(\|\cdot\|_V\) are equivalent.
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Let \(A:\ell^\infty\to\ell^\infty\) be the function \(A((a_1,a_2,a_3,\dots)) = (0,a_1,a_2,\dots)\).
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Show that A is a bounded linear operator
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Find a bounded, linear operator \(B:\ell^\infty\to\ell^\infty\) such that \(BA=I\) (that is, B is a left inverse of A)
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Show that \(AB\neq I\) (that is, B is not a right inverse of A)
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Section 5.5: Exercises 1, 2
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Thurs 01.04
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??????????????Section 5.5: Exercises 3, 4, 5, 6
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