Jumpstart - Questions and Answers Week 2

Question 1 (Greg Zitelli, Jumpstart TA Fall 2018)

The expression $$ d_{max}(f,g) = \max\{|f(x)-g(x)|:x \in M\} $$ defines a metric on the space $ \operatorname{C}(M)$ of continuous functions on a compact metric space $M$. Can a similar expression be used to define a metric on the space of continuous functions when $M$ is not compact, for instance $ \operatorname{C}(\mathbb{R})$? What about trying to define a metric on a subspace of $ \operatorname{C}(\mathbb{R})$ or on a larger space?

We could attempt to use something like $$ d_{sup}(f,g) = \sup_{x \in\mathbb{R}}|f(x) - g(x)|. $$ Unfortunately this runs into the problem of two functions being infinitly distant from each other; $f(x) = x$ and $g(x) = -x$ say. That would violate our current definition of a metric mapping $d:X\times X \to \mathbb{R}$ as we would need to go into the extended reals. Perhaps we could use only elements of $ \operatorname{C}(\mathbb{R})$ with compact non-zero sets, or rather the closure of their non-zero sets is compact, denoted by $ \operatorname{C}_c(\mathbb{R})$. While $d_{sup}$ would define a distane on this latter space, it would not be complete. To see this take the function $$ f(x)=\frac{1}{1+x^2}\text{ for }x\in \mathbb{R} $$ and show that a Cauchy sequence can be constructed in $\operatorname{C}_c(\mathbb{R})$ that converges to it (see next question if you have trouble performing the construction). The point is that the distance $d_{sup}$ does not allow to control the support during convergence.
Now, if we added the condition that the functions be bounded, the distance function would be always defined and finite. This amounts to considering the smaller space $$ \operatorname{BC}(\mathbb{R})=\big\{ f\in \operatorname{C}(\mathbb{R})\, \big |\, \sup _{x\in\mathbb{R}} |f(x)|<\infty\big\}. $$ Show that this is the case and that this space is complete with respect to the metric $d_{sup}$. On the other hand, it is also possible to give up continuity and consider the larger space $$ \operatorname{B}(\mathbb{R})=\big\{ f:\mathbb{R}\to \mathbb{R}\, \big |\, \sup _{x\in\mathbb{R}} |f(x)|<\infty\big\}, $$ on which $d_{sup}$ is well-defined and a metric (check!). Is this space complete?

Question 2

What is the completion of $ \operatorname{C}_c(\mathbb{R})$, the space of continuous functions with compact support, with metric given by $$ d(f,g)=\sup_{x\in \mathbb{R}}|f(x)-g(x)|,\: f,g\in \operatorname{C}_c(\mathbb{R})? $$ Observe that the support $\operatorname{supp}(f)$ of a function $f$ is defined by $\overline{\{ x:f(x)\neq 0\}}$. What if $ \mathbb{R}$ were replaced by a metric space $(X,d_X)$?

I can name the completion of $\operatorname{C}_{c}(\mathbb{R})$ but the proof may have some mistake (there is no longer any mistake in this adaptation). Define $C^*(\mathbb{R})$ a the space of all continuous functions which tend to 0 as $x$ tend to positive or negative infinity, that is, $$ \big\{ f\in \operatorname{C}(\mathbb{R})\, \big |\, \forall\: \varepsilon>0\:\exists\: N\in \mathbb{N} \text{ s.t. } |f(x)|\leq \varepsilon,\: |x|\geq N\big\} $$ Then $C^*(\mathbb{R})$ is the completion of $C_c(\mathbb{R})$. To see this we need to consider Cauchy sequences of functions in $C_c(\mathbb{R})$ and show that they can each be identified with an element of $C^*(\mathbb{R})$. Given such a Cauchy sequence $(f_n)_{n\in \mathbb{N}}$ and any $N\in \mathbb{N}$, we can find a limit (with respect to uniform convergence) $f^N_\infty$ for the sequence $f_n\big |_{[-N,N]}$. It is easy to verify that $$ f_N\big |_{[-M,M]}=f_M\text{ whenever }N\geq M. $$ This is due to the fact that $(f_n\big |_{[-N,N]})_{n\in\mathbb{N}}$ is a Cauchy sequence in $ \operatorname{C}\bigl( [-N,N]\bigr) $ for any $N\in \mathbb{N}$ and $[-N,N]$ is compact. Finally define $$ f_\infty(x)=f_\infty ^N(x)\text{ if }x\in [-N,N] $$ and observe that $f_n\to f_\infty$ uniformly and that $f\in \operatorname{C}^*(\mathbb{R})$ (verify!). This gives an inclusion. As for the other one, let $f\in \operatorname{C}^*(\mathbb{R})$ and let us find a Cauchy sequence in $C_c(\mathbb{R})$ that converges to it. To this end take $\varphi_n$ given by $$ \varphi _n(x)=\begin{cases} 1, &|x|\leq n,\\ 1-(|x|-n), &n<|x|\leq n+1,\\ 0, &|x|> n, \end{cases} $$ and define $f_n=f\, \varphi_n\in C_c(\mathbb{R})$, $n\in\mathbb{N}$, and show that this yields a Cauchy sequence with the desired properties which converges to $f$.

Question 3 (James Shade, incoming Fall 2018)

I was thinking about graphs as metric spaces, and this got me thinking about why we define metric spaces as we do. A finite, connected, undirected graph is certainly a metric space when the metric is the minimum number of edges required to traverse from one point to the other. However, what if we remove one of these properties? For instance, if the graph is disconnected, the only way I see to apply a concept of distance to the graph is to introduce infinite distances between disconnected points, which would preclude this from being a metric space. Similarly, if we make the graph directed, then we may lose the symmetry property (and may once again not have any paths from one point to another). This exercise got me thinking about the inclusion of symmetry and finiteness in the definition of the distance function. Suppose we were to eliminate symmetry from the definition of a metric. Does this introduce problems in establishing convergence of a sequence? In other words, can there exist a sequence $x_n$ such that $\lim_{n \to \infty}d(x_n, x) = 0$ but $\lim_{n \to \infty}d(x,x_n) \neq 0$, or vice versa? What about introducing infinite distances? Does introducing infinite distances cause any complications in how we work with these spaces?

The example of a graph is a nice one and, in fact, finding as many examples of non-trivial metric spaces as you can really helps the intuition. To your questions: on the one hand, math is a game and you can make whatever rules you wish and play according to them. So, by all means, toy with the idea of removing, replacing, and modifying conditions and study the consequences! On the other hand, there are often one or more applications that motivate the concepts we introduce. Related to your example: consider the space consisting of the points of a circle and assume that it is only possible to move along the circle in counterclockwise direction (to avoid traffic blockages). What would be a natural definition of distance in this world? Would it be symmetric? Would it be useless? How would convergence look like? Or consider the surface of the earth and define a distance between points which takes into account the difference in their elevation as well, thus accounting for the fact that going up is "harder" than going down: would such a distance be symmetric? Would convergence in this case be impacted by the asymmetric definition of distance? In all these examples, picture the typical open ball around a point.