1.1.1. Iterated triangle inequality.
Hint.
Repeatedly apply the triangle inequality, for instance using induction.
Appendix D Hints for exercises
Here we collect the hints to all of the exercises in the lecture notes, including many exercises which will not be assigned. It is only really useful if you are reading a pdf version ‘with hints in appendix’.
1 Fundamental Concepts
1.1 Metrics, Norms and Inner Products
Exercises
1.1.2. Spaces with three points.
Hint.
The main thing to check is the triangle inequality. Since \(X\) only has three elements, this can be done by brute force.
1.1.5. (PS2) Some norms on \(\R^2\).
1.1.5.a
Hint.
Use the triangle inequality for \(\abs\blank\) and regroup terms.
1.1.5.b
Hint.
This one is slightly trickier. In addition to the triangle inequality for \(\abs\blank\text{,}\) the obvious inequalities \(\abs{x_1}, \abs{x_2} \le \n
x_\infty\) are also useful. I would advise against breaking into different cases based on whether \(\abs{x_1}\) or \(\abs{x_2}\) is larger and so on.
1.1.5.c
Hint 1.
It may be helpful to look back to your sketches from the previous two parts.
Hint 2.
For the third inequality, interpret \(\n x_1\) as the inner product of the vectors \((\abs{x_1},\abs{x_2})\) and \((1,1)\text{,}\) and use the Cauchy–Schwarz inequality.
1.1.7. Calculating supremum norms.
1.1.7.b
Hint.
Don’t forget about the absolute values! Also, you will need to use basic calculus.
1.1.8. (PS2) Estimating supremum norms.
1.1.8.a
Hint.
Estimate each factor separately.
1.1.8.b
Hint.
Estimate each term separately. Remember that the definition of \(\n\blank_{\sup}\) involves not only a supremum but an absolute value.
1.1.9. (PS2) Product spaces.
Hint 1.
In one step it is helpful to use the Cauchy–Schwarz inequality in \(\R^2\text{,}\) which implies that \(ab + cd \le \sqrt{a^2 + c^2} \sqrt{b^2 + d^2}\) for all \(a, b, c, d \in \R\text{.}\)
Hint 2.
If you are struggling with this problem, one strategy is to break things down into stages:
- First, carefully write down what it would mean for \((X \times Y, d_{X \times Y})\) to be a metric space.
- Then use the definition of \(d_{X \times Y}\) to rephrase the axioms to be proved in terms of the metrics \(d_X\) and \(d_Y\text{.}\)
- Finally, try prove these rephrased versions.
In my experience marking this question over the years, many students run into problems because they effectively jump straight to this last step but have made serious errors in the first two which prevent this from working.
1.1.10. Normed spaces are unbounded.
Hint.
Consider points of the form \(\alpha x_0\) where \(\alpha \in \R\text{.}\)
1.2 Open and Closed Sets
Exercises
1.2.1. (PS3) Inclusions of balls.
1.2.1.b
Hint.
To simply things, search for a counterexample with \(x_0=y_0\text{.}\)
1.2.2. (PS3) Open sets in metric subspaces.
1.2.2.a
Hint.
Write down a detailed definition of the balls \(B_r(y)\) and \(B_r'(y)\text{.}\)
1.2.2.b
Hint 1.
Hint 2.
The reverse implication is not so bad once you unpack the definitions, but the forward direction requires some more thought. Given an open subset \(V\) of \((Y,d')\) you need to ‘cook up’ and appropriate open subset \(U\) of \((X,d)\text{.}\) Here you can look at the proof of Corollary 1.27 for some inspiration.
1.2.2.c
Hint.
While it is not so hard to argue directly, using Lemma 1.28 is much faster.
1.3 Convergence and Completeness
Exercises
1.3.1. (PS3) Some basic limits.
1.3.1.a
Hint.
Use the triangle inequality to show \(d(x_0,y_0) \le d(x_0,x_n) +
d(x_n,y_n) + d(y_n,y_0)\text{,}\) as well as the same inequality with the roles of \(0\) and \(n\) reversed. Then use this to estimate the difference \(\abs{d(x_n,y_n) - d(x_0,y_0)}\text{.}\)
1.3.1.b
Hint.
For the first part, find a way to express \(\n{x_n}\) in terms of the metric \(d\text{.}\) For the other two parts you will want to use the triangle inequality, perhaps after adding and subtracting an appropriate ‘cross term’.
1.3.1.c
Hint.
Again, experiment with adding and subtracting an appropriate ‘cross term’. The Cauchy–Schwarz inequality is also useful.
1.3.3. (PS4) Cauchy implies bounded.
Hint.
Apply the definition of \(\seq xn\) being Cauchy with \(\varepsilon=1\) to find an appropriate \(N\in\N\text{.}\) Then estimate \(d(x_n,x_m)\) in different ways depending on whether \(n\ge N\text{,}\) \(m \ge N\text{,}\) neither, or both.
1.3.4. (PS4) Pairs of Cauchy sequences.
Hint.
The inequality \(\abs{d(x_n,y_n) - d(x_m,y_m)} \le d(y_n,y_m) +
d(x_n,x_m)\) is quite useful here, and has already been established in Exercise 1.3.1.
1.3.5. Cauchy sequences and double limits.
1.3.5.a
Hint.
One method is to show that the sequence \((d(x_n,x_k))_{k\in \N}\) is Cauchy. This is related to Exercise 1.3.4.
1.4 Continuity
Exercises
1.4.1. (PS4) Metric is Lipschitz.
1.4.1.a
Hint.
The Reverse triangle inequality is useful here.
1.4.1.b
Hint 1.
A substantial portion of this problem is translating the statement “\(d
\maps X \times X \to \R\) is Lipschitz continuous” into an inequality to be proven. It is worth your time to do this translation carefully!
Hint 2.
The obvious inequality \(d(x_1,x_2) \le d_{X \times X}((x_1,y_1),(x_2,y_2))\) is useful here, as is the estimate used in the first part of Exercise 1.3.1.
1.4.2. (PS4) Image of closure under continuous map.
Hint.
Probably the simplest thing, given the tools available, is to argue directly using the definition of the closure in Definition 1.20 and the definition of continuity. Alternatively, one can argue in terms of convergent sequences, partly emulating the proof of Theorem 1.35.
1.5 The Space \(C_\bdd(X)\)
Exercises
1.5.1. Uniform limit of continuous maps.
Hint 1.
You have seen a proof of this when \(X=Y=\R\text{;}\) the proof here is very similar. The usual way to do it is a so-called ‘\(\varepsilon/3\)’ argument.
Hint 2.
Note that when choosing \(\delta\) in the definition of continuity for the functions \(f_n\text{,}\) the value you get will depend not only on \(\varepsilon\) (and the base point \(x_0\)) but also on the index \(n \in \N\text{.}\) Worryingly, we might even have \(\delta \to 0\) as \(n \to \infty\text{!}\) A way around this is to first choose \(N \in \N\) appropriately, and then appeal to the continuity of single function \(f_N\text{.}\)
1.7 Isometries
Exercises
1.7.1. An isometry between \(\R^2\) with two different norms.
1.7.1.a
Hint.
It may be helpful to look at your pictures from Exercise 1.1.5. Or else you can just start picking points \(x,y \in \R^2\) and checking the definition.
2 Completion of Metric Spaces
2.1 Dense Sets
Exercises
2.1.1. (PS5) Dense sets in product spaces.
Hint 1.
It is worth looking back to Exercise 1.3.2, either for inspiration or as a potential tool.
Hint 2.
If you like sequences, you could use the sequence characterisation of density in Lemma 2.3.
2.1.2. (PS5) Uniform continuity and Cauchy sequences.
Hint.
Use the uniform continuity of \(f\) first, and only then take advantage of the fact that \(\seq xn\) is Cauchy.
2.1.3. (PS5) Continuous extensions.
2.1.3.a
Hint.
Don’t overthink it.
2.1.3.b
Hint.
Again, don’t overthink it.
2.1.3.e
Hint 1.
You’re looking for an \(g\) where any extension \(f\) is necessarily discontinuous.
Hint 2.
Find a function which misbehaves at a point in \([a,b] \without \Q\) but is otherwise continuous \([a,b] \to \R\text{,}\) and then let \(g\) be the restriction of this function to \([a,b] \cap \Q\text{.}\)
2.1.4. (PS6) \(\C\) is separable.
2.1.4.a
Hint.
One way to start is to write down what it means for a set not to be dense, in the same way that in Exercise B.1 you wrote down what it means for a sequence not to converge. If you are having trouble accurately negating statements with quantifiers please come talk to me.
2.1.4.b
Hint 1.
If you’re having trouble thinking of a good candidate for a countable dense subset, take a look through Section 1.8, particularly Example 1.67 and Lemma 1.70.
Hint 2.
Once you’ve got your candidate for the countable dense subset, just argue directly using Definition 2.1. Alternatively, you could think about how \(\C\) is isometric to \(\R^2 = \R \times \R\text{,}\) and try to use Exercise 2.1.1.
2.2 Existence of the Completion
Exercises
2.2.1. (PS6) Equivalence relation in the proof of Theorem 2.10.
Hint.
Don’t forget that many of your tricks from Analysis 1 for manipulating limits require you to know ahead of time that some of the limits involved actually exist. Exercise 1.3.4 and Exercise 2.2.2 are be helpful here.
2.2.2. (PS5) Triangle inequality in Theorem 2.10.
Hint.
Don’t forget that many of your tricks from Analysis 1 for manipulating limits require you to know ahead of time that some of the limits involved actually exist. Exercise 1.3.4 may be helpful here.
2.2.3. (PS5) Completion vs. closure.
Hint.
Use Theorem 1.23 and Theorem 1.41.
3 Some Function Spaces
3.1 \(C^k([a,b])\)
Exercises
3.1.1. (PS6) Calculating and estimating \(C^k\) norms.
3.1.1.a
Hint.
Half of the work is already done for you in Exercise 1.1.7.
3.1.1.b
Hint.
Half of the work is already done for you in Exercise 1.1.8.
3.1.2. Uniform continuity of derivatives.
Hint.
Lemma 2.4 may be useful.
3.2 Hölder Spaces
Exercises
3.2.2. (PS7) Regularity of \(x \mapsto x^\alpha\).
3.2.2.a
Hint.
Show that there is no \(g \in C^0([0,1])\) with \(g(t)=f'(t)\) for \(t \in (0,1)\text{.}\)
3.2.2.b
Hint.
Estimate the supremum in (3.1) from below by setting \(y=0\text{.}\)
3.2.2.c
Hint.
First, use calculus to show that
\begin{gather}
(t+1)^\alpha-t^\alpha \le 1 \text{ for } t \ge 0\text{.}\tag{✶}
\end{gather}
\begin{equation*}
\frac{x^\alpha-y^\alpha}{(x-y)^\alpha} \le 1\text{.}
\end{equation*}
3.2.3. (PS7) \(C^1\) as a subset of \(C^{0,1}\).
3.2.3.a
Hint.
As you can easily check yourself, \(C^1([a,b])\) is closed under vector space operations, and so the main thing to prove here is that it is a subset of \(C^{0,1}([a,b])\text{.}\) To prove this, fix \(x,y \in [a,b]\) with \(x \ne y\text{,}\) and try to estimate the quotient in the definition of \([f]_{C^{0,1}([a,b])}\) in terms of \(f'\) using the mean value theorem. Taking a supremum, conclude that \([f]_{C^{0,1}([a,b])} \le \n
{f'}_{C^0([a,b])}\text{.}\)
3.2.3.b
Hint.
In the previous part you have hopefully already shown that \(\n{f'}_{C^0([a,b])} \ge [f]_{C^{0,1}([a,b])}\) for any \(f \in
C^1([a,b])\text{,}\) and so it suffices to show the reverse inequality. Fix \(x \in (a,b)\text{,}\) and write \(\abs{f'(x)}\) as a limit as \(y \to x\text{,}\) and then estimate inside the limit.
3.2.3.d
Hint.
Part c is useful here, as is the completeness of \(C^1([a,b])\) when equipped with the usual \(\n\blank_{C^1([a,b])}\) norm.
3.2.4. \(C^{0,\alpha}\) is non-separable.
3.2.4.a
Hint.
This is probably the hardest part of this problem. If you are getting stuck, I would recommend moving on to the others parts and doing them first. One way to compute the limit is to compare some of the terms to the limit definition of the derivative \(f_s'(t)\text{.}\)
3.2.5. (PS8) \(C^{1,\alpha}([a,b])\).
Hint 1.
Theorem 3.2 and Theorem 3.5 are very useful here. You might also want to look at their proofs for some general inspiration.
Hint 2.
Suppose that \(\seq fn\) is Cauchy in \(C^{1,\alpha}([a,b])\text{.}\) Argue that \(\seq fn\) is also Cauchy in \(C^1([a,b])\) and that \(\seq{f'}n\) is Cauchy in \(C^{0,\alpha}([a,b])\text{.}\) Now appeal to the completeness of \(C^1([a,b])\) and \(C^{0,\alpha}([a,b])\text{.}\)
Hint 3.
Suppose that \((X,d_X)\) and \((Y,d_Y)\) are metric spaces with \(Y \subset
X\text{.}\) Then a sequence \(\seq yn\) in \(Y\) is also a sequence in \(X\text{.}\) Without knowing how the metrics \(d_X,d_Y\) compare, however, we can conclude nothing about how \(\seq yn\) being Cauchy/convergent as a sequence in \((Y,d_Y)\) is related to it being Cauchy/convergent as a sequence in \((X,d_X)\text{.}\)
3.2.6. (PS7) Inclusions between Hölder spaces.
Hint.
Write \(\abs{x-y}^\beta=\abs{x-y}^{\beta-\alpha} \abs{x-y}^\alpha\text{.}\)
3.3 Integration Review
Exercises
3.3.1. (PS6) Integrals involving piecewise-linear functions.
3.3.1.b
Hint.
I strongly recommend sketching the graph of \(f\text{.}\)
3.3.1.c
Hint 1.
Seriously, do not explicitly calculate the integrals. Instead, take advantage of the inequalities you have already shown in the previous part.
Hint 2.
Use Theorem 3.7 to split up the integrals into several pieces, and then estimate the different pieces using Corollary 3.8 and Part b.
3.4 \(L^2([a,b])\)
Exercises
3.4.1. \(L^2\) inner product on \(C^0\).
Hint.
3.4.2. (PS7) Calculating and estimating \(L^2\) norms.
3.4.2.a
Hint.
To avoid having to write too many square roots, it is convenient to first calculate \(\n{g_c}_{L^2([-1,1])}^2\) and then take a square root.
4 Compact Sets
4.1 Sequential Compactness
Exercises
4.1.1. (PS8) Compactness and minimisation.
4.1.1.a
Hint 1.
Exercise B.2 is useful here.
Hint 2.
Note that question asks you to consider the possibility that \(\inf_{x \in X} f(x) = -\infty\text{.}\)
4.1.1.b
Hint.
Argue that the minimising sequence from the previous part has a convergent subsequence.
4.1.2. Bounding functions on compact spaces.
Hint.
Try an argument by contradiction.
4.1.3. Weierstrass extreme value theorem.
Hint.
Use Exercise 4.1.1.
4.1.4. Nested compact sets.
Hint 1.
For each \(n \in \N\text{,}\) pick a point \(x_n \in C_n\text{.}\) Argue that \(\seq xn\) is then a sequence in the sequentially compact set \(C_1\text{.}\)
Hint 2.
Theorem 4.5 is useful.
4.2 Total Boundedness
Exercises
4.2.1. Total boundedness for subsets.
Hint.
Part a of Exercise 1.2.2 is useful here.
4.3 Compactness
Exercises
4.3.1. (PS9) Direct proof of non-compactness.
Hint.
If you’re feeling stuck, examples from the lectures, lecture notes and problems classes may be useful inspiration.
4.3.2. Compactness and metric subspaces.
Hint.
Use Exercise 1.2.2.
4.3.3. (PS9) Continuity on compact spaces.
Hint 1.
This is relatively big hint. The official solution uses Definition 4.16 directly. The general idea is to construct an open cover in the following way. Fix \(\varepsilon \gt 0\text{.}\) Since \(f\) is continuous, for all \(x \in X\) there exists a \(\delta(x) \gt 0\) such that \(d_X(x,x') \lt \delta(x)\) implies that \(d_Y(f(x),f(x')) \lt \varepsilon\text{.}\) Then the set of all balls of the form \(B_{\delta(x)}(x)\) forms an open cover of \(X\text{,}\) and so using compactness we can find a finite subcover.
This is only the general idea, though, and you will likely find that you need to do things like fiddle around with \(\varepsilon\) versus \(\varepsilon/2\text{,}\) and also with the radii of the balls in the open cover.
Hint 2.
Alternatively, thanks to Theorem 4.22, we can argue instead using sequential continuity. For this, you will want to write down what it would mean for \(f\) not to be uniformly continuous, and then use that to construct sequences with a certain property. Sequential compactness will let you find convergent subsequences, which will then lead to a contradiction with the (pointwise) continuity of \(f\text{.}\)
4.3.4. (PS9) Images of compact sets.
4.3.4.a
Hint.
Given a sequence in \(f(X)\text{,}\) relate it to a sequence in \(X\text{,}\) and then use sequential compactness.
4.3.4.b
Hint.
Look at the definition of continuity (Definition 1.42) and rewrite the relevant inequalities in terms of points being in certain balls.
4.3.4.c
Hint.
Part b is useful here.
4.3.5. (PS8) A closed and bounded set which is not compact.
4.3.5.b
Hint.
To get some intuition, draw graphs for small values of \(n\) and \(m\text{.}\) Since you only need a lower bound, it suffices to find a point \(t \in [0,\pi]\) where \(\abs{f_n(t)-f_m(t)} \ge 1\text{.}\)
4.3.5.c
Hint.
Theorem 1.37 is useful.
4.3.5.d
Hint 1.
Suppose that \(r \in (0,1)\) and let \(f \in F\text{.}\) What can you say about the open ball \(B_r(f)\text{?}\)
Hint 2.
4.3.5.e
Hint.
Same hints as for the previous part.
4.3.5.f
Hint 1.
Once again, (✶) is useful.
Hint 2.
It is tempting to think that the only sequences in \(F\) are subsequences of \(\seq fn\text{,}\) but this is not quite true. For instance, the constant sequence \((f_1,f_1,\ldots)\) is a sequence in \(F\text{,}\) but it is certainly not a subsequence of \(\seq fn\text{.}\)
4.4 Relative Compactness
Exercises
4.4.1. (PS9) Relative compactness and subsequences.
Hint.
Showing that Part i implies Part ii is not so bad, but the reverse implication requires a bit more ingenuity. One way to proceed, given a sequence \(\seq xn\) in \(\overline S\text{,}\) is to use the definition of the closure to argue that there is another sequence \(\seq sn\) in \(S\) with \(d(x_n,s_n) \lt 1/n\text{.}\)
4.5 The Arzelà–Ascoli Theorem
Exercises
4.5.1. (PS9) Finite sets are equicontinuous.
Hint.
Let \(x_0 \in X\) and \(\varepsilon \gt 0\text{.}\) Use continuity of each \(f_n\) at \(x_0\) to find a \(\delta_n \gt 0\text{.}\) Now choose \(\delta \gt 0\) appropriately in terms of \(\delta_1,\ldots,\delta_n\text{.}\)
4.5.2. (PS10) Bounded subsets of \(C^{0,\alpha}\) relatively compact in \(C^0\).
4.5.2.a
Hint.
This is easy. Since \(C^0([a,b])\) is both a metric space and a normed space, \(S\) being bounded can be defined using either the first or the last bullet point in Definition 1.15.
4.5.2.b
Hint 1.
Make sure that your proof that \(S\) is equicontinuous cannot be mistaken for a ‘proof’ that every \(f \in S\) is continuous.
Hint 2.
Let \(f \in S\) and \(x,x_0 \in [a,b]\) with \(x \ne x_0\text{.}\) Now estimate \(\abs{f(x)-f(x_0)}\) in terms of \([f]_{C^{0,\alpha}([a,b])}\text{.}\) Show that, since \(S\) is bounded in \(C^{0,\alpha}([a,b])\text{,}\) this implies equicontinuity.
4.5.3. (PS10) A set which is not equicontinuous.
4.5.3.a
Hint 1.
If you are at all uncertain about your ability to correctly negate the statement ‘\(F\) is equicontinuous’ on the fly, then work out the negation very carefully on scratch paper beforehand.
Hint 2.
For instance, you can take \(x_0=0\) and \(\varepsilon = 1/2\text{.}\) It may help to sketch a picture.
4.5.4. (PS10) Compactness and indefinite integrals.
Hint 1.
Remember that when we say that a subset of a metric space is bounded, we always mean this in the sense of Definition 1.15.
Hint 2.
Show that \(\Phi(F)\) is bounded in \(C^{0,1}([a,b])\) and use Corollary 4.31. Another possibility is to show that \(\Phi(F)\) is bounded in \(C^1([a,b])\) and then apply Exercise 3.2.3 and Corollary 4.31.
5 Uniform Approximation
5.1 The Weierstrass Approximation Theorem
Exercises
5.1.1. (PS10) \(C^0([a,b])\) is separable.
5.1.1.a
Hint.
This is relatively big hint. Let \(p \in P\text{,}\) say
\begin{equation*}
p(x) = \alpha_0 + \alpha_1 x + \cdots + \alpha_M x^M
\end{equation*}
for some \(M \in \N\) and coefficients \(\alpha_0,\ldots,\alpha_M \in \R\text{.}\) Argue that we can find a sequence \(\seq qn\) in \(Q\) of polynomials of the form
\begin{gather*}
q_n(x) = \beta_0^{(n)} + \beta_1^{(n)} x + \cdots + \beta_M^{(n)} x^M
\end{gather*}
whose coefficients converge to the coefficients of \(p\text{,}\) i.e. \(\beta_m^{(n)} \to \alpha_m\) as \(n \to \infty\) for all \(m \in \{1,\ldots,M\}\text{.}\) Then show that this implies \(q_n \to p\) in \(C^0([a,b])\text{.}\)
5.1.1.b
Hint.
This follows relatively easily from the previous part and the Weierstrass approximation theorem.
5.1.1.c
Hint 1.
This is probably the most sophisticated cardinality argument in the entire unit, and so if you are pressed for time you may just want to skip it and wait for the solutions. On the other hand it is doable with only the tools in Section 1.8; see the next hint.
Hint 2.
Let \(Q_n\) denote the set of polynomials in \(Q\) of degree \(\le
n\text{.}\) First argue that \(Q_n\) is countable by appealing to Lemma 1.70. Then notice that \(Q = \bigcup_{n\in\N} Q_n\) and appeal to Lemma 1.72.
5.1.2. Trigonometric polynomials dense in \(L^2([a,b])\).
5.1.2.a
Hint.
Fix \(f \in C^0([0,2\pi])\text{.}\) Now let \(f_n\) be a continuous function obtained by replacing the graph of \(f\) for \(x \in [0,1/n]\) with a straight line segment in such a way that \(f_n \in A\text{.}\) Estimate \(\n{f-f_n}_{L^2(0,2\pi)}^2\text{,}\) and show that it tends to \(0\) as \(n
\to \infty\text{.}\)
5.1.2.b
Hint.
Given \(f \in A\text{,}\) extend it to \(2\pi\)-periodic continuous function \(\tilde f \maps \R \to \R\) and apply Theorem 5.5 to find a trigonometric polynomial \(p \in T\) which is close to \(f\) in terms of \(\n\blank_{\sup}\text{.}\) Now use this to estimate \(\n{f-p}_{L^2(0,2\pi)}\text{.}\) Finally, tie things together by applying the previous part.
5.2 The Stone–Weierstrass Theorem
Exercises
5.2.1. (PS11) Some non-examples.
5.2.1.a
Hint.
It is enough to look at polynomials with degree \(1\text{.}\)
5.2.1.b
Hint.
Consider, say, the function \(f \in C^0([0,1])\) defined by \(f(x)=x\text{.}\) Show that \(f \notin \overline S\) by either
- finding a uniform lower bound on \(\n{f-c}_{C^0([0,1])}\) for \(c \in S\text{;}\) or
- assuming that \(\seq sn\) is a sequence in \(S\) converging to \(f\) and deriving a contradiction.
5.2.1.c
Hint.
Note that \(S\) being an algebra does not immediately follow from \(P([0,1])\) being an algebra; it requires proof. To show that \(S\) separates points, it is enough to consider the polynomial \(p \in
S\) defined by \(p(x)=x\text{.}\)
5.2.2. Lattice operations are continuous.
5.2.2.a
Hint.
Directly verifying the \(\varepsilon\)–\(\delta\) definition of continuity is doable but a bit of a pain. By Exercise 1.3.2 and Theorem 1.46, it is enough to instead show that for any two sequences \(\seq fn\) and \(\seq gn\) in \(C_\bdd(X)\) with \(f_n \to f_0\) and \(g_n \to g_0\text{,}\) we have \(f_n g_n \to f_0
g_0\text{.}\)
5.2.2.b
Hint.
One approach is to fix \(x \in X\) and then estimate by breaking things up into a bunch of cases, eventually obtaining a bound which doesn’t depend on \(x\) and taking a supremum. It’s perhaps a bit easier, though, to use the same sort of sequence argument as in the hint to the previous part combined with (5.4) and (5.5).
5.2.5. (PS11) Separating points.
5.2.5.a
Hint.
First write down what it means for \(S\) to not separate points. Then use the fact that \(f \in \overline S\) implies that there exists a sequence \(\seq fn\) in \(S\) with \(f_n \to f\text{.}\)
5.2.5.b
Hint.
Use the previous part to show that the identity function \(t \mapsto
t\) does not lie in \(\overline S\text{.}\)
5.2.6. A convergence criterion for series.
Hint.
\begin{equation*}
a_{N + m} = \frac{a_{N + m}}{a_{N + m - 1}} \frac{a_{N + m - 1}}{a_{N + m - 2}} \cdots \frac{a_{N + 1}}{a_N} a_N
\end{equation*}
for every \(m \in \N\text{.}\)
5.2.7. Another convergence criterion for series.
5.2.7.a
Hint.
Show that the function \(g \maps (0, \infty) \to \R\text{,}\) \(x \mapsto
x^\beta\text{,}\) is convex on \((0, \infty)\text{.}\) Conclude that \(g(x) \ge g(1)
+ (x - 1) g'(1)\) for all \(x \in (0, \infty)\text{.}\)
6 The Baire Category Theorem
6.1 The Theorem
Exercises
6.1.3. Interiors, closures, and complements.
6.1.3.b
Hint.
Show that \(X \without (\overline{X \without S})\) is an open set contained in \(S\text{,}\) while \(X \without S^\circ\) is a closed set containing \(X \without S\text{.}\)
6.1.4. (PS11) Baire for linear subspaces.
6.1.4.a
Hint.
Let \(x,y \in \overline L\) and \(\alpha \in \R\text{.}\) Then there are sequences \(\seq xn\) and \(\seq yn\) in \(L\) converging to \(x\) and \(y\text{.}\) Argue that \((x_n+\alpha y_n)_{n\in\N}\) is a sequence in \(L\) converging to \(x+\alpha y\text{,}\) and hence that \(x+\alpha y \in
\overline L\text{.}\)
6.1.4.b
Hint.
Given \(x \in L^\circ\text{,}\) there exists \(r \gt 0\) such that \(B_r(x) \subseteq
L\text{.}\) To see that \(B_r(0) \subseteq L\) as well, argue that any \(y \in
B_r(0)\) can be written as \(y=z-x\) where \(z \in B_r(x)\text{.}\) Finally, to see that \(L=X\text{,}\) pick an arbitrary \(x \in L\) and choose \(\alpha \in \R\) so that \(\alpha x \in B_r(0)\text{.}\)
6.1.4.c
Hint.
6.1.4.d
Hint.
Use Corollary 6.6 and the previous part.
6.1.5. A fact about \(\Q\).
Hint.
Argue by contradiction. Suppose that \(\Q = \bigcap_{n \in \N} U_n\text{,}\) where \(U_n \subseteq \R\) is open for every \(n \in \N\text{.}\) Suppose that \((q_n)_{n \in \N}\) is an enumeration of \(\Q\text{.}\) Then \(\R =
\bigcup_{n\in\N} (\R \without U_n \cup \{q_n\})\text{.}\) Show that this contradicts the Baire category theorem.)
