Math 104 Assignment 12

1. Uniform convergence and boundedness

   Suppose that $X$ is a set and $(f_n)_n$ is a sequence of functions $X \to \R$, so that each $f_n$ is bounded. Suppose further that $(f_n)_n$ converges uniformly to some $f : X \to \R$. Show that $(f_n)_n$ is uniformly bounded: that is, there is some $T \in \R$ so that for all $x \in X$ and all $n \in \N$, \[\abs{f_n(x)} \leq T.\]

2. Uniform convergence and sums

   In this problem, we will make the assumption that sequences are indexed beginning at $0$ rather than at $1$.

   1. Suppose that $(a_k)_k$ is a sequence in $\R$. For each $n \in \N$, set \[s_n = \sum_{k=0}^n a_k.\] If $(s_n)_n$ converges, we write $\sum_{k=0}^\infty a_k = \lim_{n\to\infty}s_n.$ Show that the sequence $(s_n)_n$ converges if and only if for every $\epsilon \gt 0$ there is some $N \in \N$ so that for any $n, m \in \N$ with $N \lt n \lt m$, \[\abs{\sum_{k=n}^m a_k} \lt \epsilon.\]
   2. Suppose now that $X$ is a set, and $(f_k)_k$ is a sequence of function $X \to \R$. For each $n \in \N$, set \[g_n = \sum_{k=0}^n f_k.\] Show that the sequence $(g_n)_n$ converges uniformly if and only if for every $\epsilon \gt 0$ there is some $N \in \N$ so that for any $n, m \in \N$ with $N \lt n \lt m$, \[\sup_{x \in X}\abs{\sum_{k=n}^m f_k(x)} \lt \epsilon.\] In this case, we say that $\sum_{k=0}^\infty f_k$ converges uniformly, and use this to denote the limit of $(g_n)_n$.
   3. Suppose that $(B_k)_k$ is a sequence in $\R_{\geq 0}$ so that \[\sum_{k=0}^\infty B_k\] converges, and $(f_k)_k$ is a sequence of functions $X \to \R$ so that $f_k$ is bounded by $B_k$: that is, $\abs{f_k(x)} \leq B_k$ for all $x \in X$. Use parts A and B to show that \[\sum_{k=0}^\infty f_k\] converges uniformly.
   4. Show that if $\abs{r} \lt 1$, then \[\sum_{k=0}^\infty r^k = \frac1{1-r}.\] (Hint: first notice that $(1-r)(1+r+r^2+\cdots+r^n) = 1-r^{n+1}$.)
   5. Suppose that $(a_k)_k$ is a sequence in $\R$ so that $\limsup_{k\to\infty} \abs{a_k}^{1/k} \lt 1$. Prove that \[\sum_{k=0}^\infty a_k x^k\] converges uniformly on $[-1, 1]$. (Here, "$a_kx^k$" is being used to mean the function $x \mapsto a_kx^k$.)

   A function of the form above is called a power series. Since a power series is the uniform limit of polynomials—which are continuous—it is continuous.

3. A poorly-behaved function

   Let $C : \R \to [-1, 1]$ be a continuous function with the properties that $C(2k) = 1$ and $C(2k+1) = -1$ for every $k \in \Z$, and that $\abs{C(x)-C(y)} \leq 4\abs{x-y}$ for all $x, y \in \R$.

   1. Prove that \[\sum_{{\color{red}j}=0}^\infty \paren{\frac34}^j C(9^j x)\] converges uniformly to some function $W : \R \to \R$.
   2. Let $n \in \N$ and $a \in \Z$. Show that \[\abs{W\paren{\frac{a+1}{9^n}} - W\paren{\frac{a}{9^n}}} \gt \paren{\frac34}^n.\] (Hint: the analysis here requires getting your hands dirty, and you'll probably need to make use of both 2D and its hint. Break the sum up into two pieces, and show that the tail is large enough to overcome the first piece.)
   3. Conclude that $W$ is continuous, but not differentiable at any point in $\R$.
   

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment. They are coloured by approximate difficulty: easy/medium/hard.

• Uniform convergence and derivatives

  For each $n \in \N$, let \begin{align*} f_n : \R &\longrightarrow \R \\ x &\longmapsto \frac{x}{1+nx^2}. \end{align*} Show that the sequence $(f_n)_n$ converges uniformly (on $\R$) to some function $f$, and that $(f_n')_n$ converges to $f'$ pointwise on $\R\setminus\set0$, but not at $0$.

• Pointwise convergence versus metric space convergence

  Suppose $S$ is a set, and consider the set of functions from $S$ to $\R$, $\R^S = \set{f : S \to \R}$. Let us say that a metric $d$ on $\R^S$ gives rise to the topology of pointwise convergence if whenever $(f_n)_n$ is a sequence of functions in $\R^S$ we have that $(f_n)_n$ converges to a function $f$ with respect to $d$ if and only if $(f_n)_n$ converges pointwise to $f$.

  Show that there is a metric on $\R^S$ giving rise to the topology of pointwise convergence if and only if there is a one-to-one function $\varphi : S \to \N$.

• Uniform convergence versus metric space convergence

  1. Suppose that $S$ is a set and $(M, d_M)$ is a bounded metric space. Define a metric $d_\infty$ on $M^S = \set{f : S \to M}$ by \[d_\infty(f, g) = \sup_{s \in S} d_M(f(s), f(g)).\] Show that a sequence of functions $(f_n)_n$ in $M^S$ converges uniformly to $f : S \to M$ if and only if $(f_n)_n$ converges to $f$ with respect to the metric $d_\infty$.
  2. Let $(M, d_M)$ be a metric space. Define $d_M' : M \times M \to \R_{\geq 0}$ by $d_M'(x, y) = \min(1, d_M(x, y))$. Show that $d_M'$ is a metric equivalent to $d_M$ (i.e., $d_M'$ is a metric such that sets are open with respect to $d_M$ if and only if they are open with respect to $d_M'$). Notice that $(M, d_M')$ is bounded.
  3. Suppose that $X$ is a set, and $d, d'$ are equivalent metrics on $X$. Show that sequences converge with respect to $d$ if and only if they converge with respect to $d'$.
  4. Suppose that $S$ is a set and $(M, d_M)$ is an arbitrary metric space. Show that there is a metric on $M^S$ so that sequences in $M^S$ converge with respect to the metric if and only if they converge uniformly. (We say "the topology of uniform convergence is metrizable".)

• Monotone convergence of functions (Dini's Theorem)

  Suppose that $K$ is a compact metric space, and $(f_n)_n$ is a sequence of continuous functions $K \to \R$, so that $f_1 \leq f_2 \leq f_3 \leq \cdots$. Suppose further that $(f_n)_n$ converges pointwise to some continuous function $f$: that is, for every $x \in K$, \[\limni f_n(x) = f(x).\] Prove that $(f_n)_n$ converges to $f$ uniformly.

• A norm on integrable functions

  Let $R$ be the set of integrable functions on the interval $[a, b]$. Given $f \in R$, define \[\norm{f}_2 = \paren{\int_a^b\abs{f(t)}^2\,dt}^{\frac12}.\]

  1. Show that $\norm\cdot_2$ satisfies the triangle inequality in the following sense: for all $f, g, h \in R$, \[\norm{f-h}_2 \leq \norm{f-g}_2 + \norm{g-h}_2.\]
  2. Suppose $f \in R$. Show that for any $\epsilon \gt 0$, there is a continuous function $g \in R$ with \[\norm{f-g}_2 \lt \epsilon.\] (Hint: choose a suitable partition $P = \set{x_0, \ldots, x_n}$, and take a piece-wise linear function $g$ so that $g(x_i) = f(x_i)$.)
  3. Suppose $f \in R$. Show that for any $\epsilon \gt 0$ there is a polynomial $p \in R$ so that \[\norm{f-p}_2 \lt \epsilon.\] (Hint: for this one you need Stone's Theorem, which we hopefully will cover next week.)

  The function $d_2(f, g) = \norm{f-g}_2$ on $R\times R$ is a pseudo-metric on $R$: it satisfies the triangle inequality, $d_2(f,f) = 0$, and $d_2(f, g) = d_2(g, f)$, but $d_2(f, g)=0$ does not imply $f = g$. If we define $\sim$ by $f\sim g$ whenever $d_2(f, g) = 0$, then $R/\sim$ becomes a metric space; what we have done is show that (equivalence classes of) polynomials are dense in $R/\sim$.

• Monotony

  State precisely what is meant by:

  • a monotone sequence of functions;
  • a sequence of monotone functions; and
  • a monotone sequence of monotone functions.

  Provide examples to show that these three concepts are different.

• Uniform closures of algebras need not be algebras

  Recall that for $x \in \R$, $\floor{x}$ (the floor of $x$) is the greatest integer $k \in \Z$ with $k \leq x$. For $n \in \N$, define \begin{align*} \varphi_n: \R &\longmapsto \R\\ t & \longmapsto 2^{-n}\floor{2^nt}. \end{align*} Finally, let $\mathscr{A}$ be the algebra of functions generated by $(\varphi_n)_n$. That is, $\mathscr{A}$ consists of all functions which are finite sums of scalar multiples of finite products of $\varphi_n$'s.

  Let $\overline{\mathscr{A}}$ be the uniform closure $\mathscr{A}$. Show that $\overline{\mathscr{A}}$ is not an algebra. (Hint: $x\mapsto x$ is in $\overline{\mathscr{A}}$, but $x \mapsto x^2$ is not.)

  Reflect on what this example says about trying to plot quadratic or linear functions on a pixelated display.

• More on power series

  Prove that a power series is differntiable, and its derivative is given by differentiating the sum term-by-term. (This is not immediate and does take work, as in general, uniform limits of differentiable functions need not be differentiable, or may have derivatives unrelated to the derivatives of their approximants.) Conclude that a power series is infinitely differentiable.

• Functions orthogonal to polynomials are zero

  Let $f : [0, 1] \to \R$ be continuous.

  1. Show that $f \equiv 0$ if \[\int_0^1 f(t)^2\,dt = 0.\]
  2. Suppose that for every $n \in \N\cup\set0$, \[\int_0^1 f(t)t^n\,dt = 0.\] Show that $f \equiv 0$. (Hint: by the end of the week, we will have proved the following theorem which you may find useful: if $K \subseteq \R$ is compact and $g : K \to \R$ is continuous, there is a sequence of polynomials $(p_n)_n$ which converges uniformly to $g$ on $K$. Use this to show that $\int_0^1 f(t)^2\,dt=0$.)

• Another weird pathology

  Show by example that there exists a function $f : \R \to \R$ such that $f$ takes on every real value on every non-empty open set. That is, for any $U \subseteq \R$ open and any $y \in \R$, there is some $x \in U$ with $f(x) = y$. (This isn't really related to this week's material in particular, it's just an interesting problem.)