MA30252: Advanced Real Analysis

In the Week 2 problems class, we talked about some different ways in which the sort of graphs in the Graphs and Networks unit can be thought of as metric spaces. Focusing on one of the simpler options, we calculated some open balls and using results from Section 1.2, and realised that all subsets in this metric space were both open and closed!

In the Week 3 problems class, we considered metric spaces of the form $(\mathbb{R},d_f)$ where $f:\mathbb{R}\to \mathbb{R}$ is an injective map and $d_f(x,y)=|f(x)-f(y)|\text{.}$ We showed that this space is isometric to $(f(\mathbb{R}),d)\text{.}$ As an example we took $f=x\mapsto e^x\text{,}$ and used this to show that $(\mathbb{R},d_f)$ was unbounded and also not complete.

In the Week 4 Problems class, we showed that the set $S=\{k/2^n:k\in \mathbb{Z},n\in \mathbb{N}\}$ is dense in $\mathbb{R}\text{.}$ To do this, we noted that for any $n\in \mathbb{N}$ and any $x\in \mathbb{R}\text{,}$ there must be at least one $k\in \mathbb{Z}$ with $x\in [k/2^n,(k+1)/2^n]$ as the union of these intervals (for a fixed $n$) is all of $\mathbb{R}\text{.}$ Thus the point $k/2^n\in S$ has $|x-k/2^n|\le 1/2^n\text{.}$ To prove density, we then go back and pick $n$ ahead of time so that $1/2^n<\epsilon \text{.}$

We then made the easy observation that supersets of dense sets are always dense, at which point the density of $S$ implies the density of $\mathbb{Q}$ as a corollary. We went on to discuss how this argument could be modified, for instance by multiplying each of the elements of $S$ by the irrational number $\sqrt{2}\text{,}$ to show that $\mathbb{R}\setminus \mathbb{Q}$ is dense in $\mathbb{R}\text{.}$

In the Week 4 Problems class, we considered the constant function $f:\mathbb{Q}\to \mathbb{R}$ defined by $f(x)=1$ and its possible extensions to a function $\hat{f}:\mathbb{R}\to \mathbb{R}\text{.}$ Clearly one such extension is the constant function $\hat{f}(x)=1\text{,}$ and by applying Lemma 2.4 we concluded that this was the only continuous extension.

In the Week 4 Problems class, we started a proof that the space $B(\mathbb{R})$ is not separable. We defined, for each $x\in \mathbb{R}\text{,}$ the function $f_x\in B(\mathbb{R})$ given by $f_x(x)=1$ and $f_x(t)=0$ for $t\ne x\text{.}$ We checked that $\Vert f_x-f_y{\Vert }_{sup}\ge 1$ for $x\ne y$ (in fact we have equality), and I waved my arms for a bit about how this makes it feel like $B(\mathbb{R})$ must be an awfully big place.

In the Week 5 Problems class, we showed how this implied that any dense subset $G\subset B(\mathbb{R})$ must have at least the cardinality of the real numbers, and hence be uncountable. This general technique is also the one outlined in Exercise 3.2.4 for Hölder spaces.

In the Week 5 Problems class, we showed that if $\alpha >1\text{,}$ then any $f\in C^0([a,b])$ with $[f{]}_{C^{0,\alpha }}<\mathrm{\infty }$ is constant. There is an urban legend about someone writing a whole thesis on these functions — moral is that you should always check that any new definition has at least one interesting example.

In the Week 5 Problems class, we showed that the set $C^0([-1,1])\setminus C^1([-1,1])$ is dense in the metric space $C^0([-1,1])\text{.}$ First we observed that the function $g=|\cdot |$ lies in this set, and then we thought about perturbing a given $f\in C^1([-1,1])$ by adding a small multiple of this function. Ultimately this was a ‘soft’ argument, in the sense that we didn’t have to do any particularly complicated estimates.

There were several other nice suggestions for how we might argue, including one using more complicated piecewise linear functions. We will revisit these sorts of approximations later in Chapter 5; see for instance Exercise 5.2.3.

In the Week 6 Problems class, we considered definite integration over a subinterval $[c,d]\subset [a,b]$ as an operation on $C^0([a,b])$ functions, and showed that this could be continuously extended to $L^2([a,b])\text{.}$ More precisely, we considered the mapping $\mathrm{\Phi }:C^0([a,b])\to \mathbb{R}$ defined by $\mathrm{\Phi }(f)={\int }_c^{d}f(t)dt\text{.}$ We decided that, if $\mathrm{\Phi }$ was Lipschitz (and hence uniformly) continuous when $C^0([a,b])$ was equipped with the metric induced by ${⟨\cdot ,\cdot ⟩}_{L^2([0,1])}\text{,}$ then by Lemma 2.4 it could be uniquely continuously extended to a mapping $\hat{\mathrm{\Phi }}:L^2([a,b])\to \mathbb{R}\text{.}$ We then showed that $\mathrm{\Phi }$ indeed had this Lipschitz continuity, where here the trick involved writing $\mathrm{\Phi }(f)={⟨1,f⟩}_{L^2([c,d])}$ and using the Cauchy–Schwarz inequality.

The general pattern here is typical of working with completions: To understand something about the completion, we first make sure we understand it for the dense subset we know better, and then try to prove some estimates which allow us to extend this understanding to the full completion.

In the Week 7 Problems class, we defined what it meant for a function $f:X\to \mathbb{R}$ from a metric space to the real numbers to be locally bounded. After thinking about some simple examples, we decided that this was, in general, a strictly weaker definition than boundedness. Finally and most importantly, we showed that if the metric space $X$ is compact, then all locally bounded functions have to be bounded. The argument we gave, which involved finding a finite subcover of a particular open cover of $X\text{,}$ is a simple example of a general strategy. We will see more complicated arguments of this type for instance in Exercise 4.3.3.

In the Week 8 Problems class, we looked at a ‘straw man’ solution to Exercise 2.1.2 which would have received few if any marks on an exam, and spent some time talking about what was wrong with it and different ways to fix it. This is related to Section G.2 and Section G.3.

In the Week 8 Problems class, we considered a certain set of piecewise linear functions in $C^0([0,1])\text{.}$ After reminding ourselves what it meant for a set of functions to be equicontinuous, and then carefully negating this, we decided that our set was probably not equicontinuous at $0\text{.}$ Then we rolled up our sleeves and worked through the $\epsilon$s and $\delta$s.

For most of the Week 9 Problems class, we walked through Exercise 3.2.5 about the definition and completeness of the space $C^{1,\alpha }([a,b])\text{,}$ taking full advantage of the completeness of the related spaces $C^1$ and $C^{0,\alpha }$ already established in lectures.

At the very end of the Week 9 Problems class, we showed that, for any $x,y\in \mathbb{R}\text{,}$ $min\{x,y\}=\frac{1}{2}(x+y)-\frac{1}{2}|x-y|$ and $max\{x,y\}=\frac{1}{2}(x+y)+\frac{1}{2}|x-y|\text{.}$ These identities will be useful in Section 5.2.

In the Week 10 Problems class, we defined what it would mean for a subset $F\subseteq C_{\mathrm{b}}(X)$ to be uniformly equicontinuous. We then showed that, when $X$ is compact, equicontinuity implies uniform equicontinuity. This ended up being incredibly similar to Exercise 4.3.3 from Problem Sheet 9, and together we came up with an argument using covers and subcovers. We started off by making some plausible guesses for which open cover to use and how to pick the relevant $\delta >0\text{,}$ but as often happens this didn’t quite work out the first time. Undeterred, we ‘went back in time’ and modified our choices so that things did work out.

In the Week 11 Problems class, we considered the set $S\subset C_{\mathrm{b}}([0,\mathrm{\infty }))$ of continuous and bounded piecewise linear functions with ‘finitely many pieces’. First we convinced ourselves that the requirement that functions in $S$ be bounded meant that their rightmost ‘piece’ had to be constant. Then we checked that all of the hypotheses of Corollary 5.9 were satisfied, except that the domain $X=[0,\mathrm{\infty })$ was not compact (since unbounded). Thus Corollary 5.9 was inconclusive. We then sketched an argument that $S$ was not dense, since for $f=\sin \in C_{\mathrm{b}}([0,\mathrm{\infty }))$ and any $g\in S\text{,}$ looking at the rightmost ‘piece’ of $g$ gave $\Vert f-g{\Vert }_{sup}\ge 1/2\text{.}$ This last part of the argument is quite similar to the second part of Exercise 5.2.1.

In the Week 11 Problems class, we considered the set $S$ of $f\in C^{0,1}([0,1])$ with $[f{]}_{C^{0,1}([0,1])}\le 1\text{,}$ i.e. with $|f(x)-f(y)|\le 1\cdot |x-y|$ for all $x,y\in [0,1]\text{.}$ We guessed that $S$ was closed as a subset of $C^0([0,1])$ because of the non-strict inequality in its definition, and then gave a proof of this using sequences. Then we quickly went over an argument that $S^{\circ }=\varnothing \text{.}$ The trick we used was to first find a sequence $(g_n{)}_{n\in \mathbb{N}}$ in $C^0([0,1])$ with $\Vert g_n{\Vert }_{C^0}\to 0$ but $[g_n{]}_{C^{0,1}}\ge 1\text{.}$ Then we considered the sequence $(f_n{)}_{n\in \mathbb{N}}$ given by $f_n=f+3g_n\text{,}$ where the constant $3$ was chosen so that $[f_n{]}_{C^{0,1}}\ge 3[g_n{]}_{C^{0,1}}-[f{]}_{C^{0,1}}\ge 2>1$ so that $f_n\notin S\text{.}$

In Week 12 Lecture 1, we did a broad overview of (most of) the definitions in the unit, organising them according to which type of object they applied to, e.g. metric space, subset of a metric space, function between metric spaces etc.. At the end we also talked about some of the ways in which our various function spaces fit together.

In Week 12 Lecture 2, we did a broad overview of (most of) the results in the unit, organised somewhat differently than in the lectures notes. Of course there wasn’t time to fully state every result; the idea is to either jog your memory and send you to the notes for the details.