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1

1.1

  • Does the sequence \(f_n(x)= \min(x/n;1)\) converge uniformly/point-wise on \([0,R)\) for any fixed \(R\gt 0\)? If it does, to what?
  • Does the sequence \(f_n(x)= \min(x/n;1)\) converge uniformly/point-wise on \([0,\infty)\)? If it does, to what?

1.2

Does \(f_n(x)=\frac{1}{1+(x-n)^{2}} \frac{1}{1+x^2}\) converge uniformly/point-wise on \(\R\)? If it does, to what?

1.3

Does \(f_n(x)=\frac{n^2}{n^2+x^{2}}\) converge uniformly/point-wise on \(\R\)? If it does, to what?

1.4

Let \(F\colon C^{0}_b(\R;\R) \to C^{0}_b(\R;\R)\) be given by

\begin{equation*} F(f)(x):= \frac{1}{1+x^2 f(x)^2} \end{equation*}

Show that \(F \colon C^{0}_b(\R;\R) \to C^{0}_b(\R;\R) \) is NOT continuous

Hint: take \(f(x)=\max(0;1-|x|)\) a traveling wave \(f_n= f(x-n)\) and consider \(g_n(x) = \frac{1}{n}f_n(x) \).

2

Read the following setup. The exercise is below

Let \(f\colon [0,1]\times [0,1]\to \R\) and define

\begin{equation*} g_{t}(x)=f(t,x) \end{equation*}

We want to show that \(t\to g_t\) is a continuous map from \([0,1]\to C^{0}([0,1];\R)\) if and only if \(f\in C^{0}([0,1]^2;\R)\). The space \(C^0([0,1];\R)\) is endowed with the uniform norm \(\|\cdot\|_{\sup}\). We make this statement precise below.

2.1

Suppose that for each \(t\), \(g_t\) is a function on \([0,1]\) that is continuous on \([0,1]\) i.e. \(g_t\in C^{0}([0,1];\R)\). Suppose furthermore that the map \(t\to g_t\) is a continuous map \([0,1]\to C^{0}([0,1];\R)\) with the latter being endowed with the uniform norm \(\|\cdot\|_{\sup}\).

Show that the function on \([0,1]\times[0,1]\) obtained by setting \(f(t,x):=g_t(x)\) is continuous on \([0,1]^2\)

2.2 ​   harder

Show that if \(f\colon [0,1]\times [0,1]\to \R\) is a continuous function then \(t\to g_t\) is a continuous map \([0,1]\to C^{0}([0,1];\R)\) where

\begin{equation*} g_{t}(x):=f(t,x) \end{equation*}

Hint: Prove first that \(f\) is uniformly continuous. Then use that \(\epsilon-\delta\) relation for \(f\) (or modulus of continuity) to show that \(\|g_t-g_{t'}\|_{\infty}\) is small if \(|t-t'|\) is small.

2.3 ​   hard

Let us construct a counterexample if the space in the \(x\) variable is not compact. Suppose \(f\colon [0,1]\times \R\to \R\) is continuous and bounded. Show that setting

\begin{equation*} g_{t}(x):=f(t,x) \end{equation*}

the function \(t\to g_t\) may fail to be a continuous map seen as a map \( [0,1] \to C^{0}(\R;\R)\).

Hint: Construct \(g_t = f(t,x)\) to be a traveling wave centered around \(x_0 = 1/t \) if \(t\gt 0 \). Find the appropriate value for \(g_0(x)\). Check that the \(f(t,x)\) thus defined is continuous.

2.4 ​   remark no_points

Notice that in all but the last statement, the space \([0,1]\) for the \(x\) variable can be replaced by any cpt metric space \(X\).

Notice that in all statements above, the time space \([0,1]\) can be replaced by \(\R^n\).

3

Let \(\alpha\in(0,1]\). A function \(f\colon X \to \R \) with \(X\) a metric space is \(\alpha\)-Hölder continuous if

\begin{equation*} |f(x)-f(y)|\leq C d(x,y)^{\alpha}\; \text{ for some } C\in[0,\infty) \text{ and } \forall x,y\in X \text{ with } d(x,y)\lt 1. \end{equation*}

From now on let us take \(X=[0,1]\).

Let us define

\begin{equation*} \begin{aligned} & \|f \|_{\dot{\mathcal{C}}^{\alpha}} := \sup_{\substack{x,y\in X\\ |x-y| \lt 1}} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}} \\ & \|f \|_{\mathcal{C}^{\alpha}} = \|f\|_{\sup}+ \|f \|_{\dot{\mathcal{C}}^{\alpha}} \end{aligned} \end{equation*}

We define the set of functions

\begin{equation*} \mathcal{C}^{\alpha}([0,1];\R)=\Big\{ f , \|f\|_{\mathcal{C}^{\alpha}}\lt \infty \Big\} \end{equation*}

If \(\alpha=1\) one writes \(\mathcal{C}^{0,1}\) instead of \(\mathcal{C}^{1}\) and such functions are called Lipschitz functions, not to be confused with \(C^1\) functions.

3.1 ​   no_points no_answer_required

Convince yourself that

\begin{equation*} \|f \|_{\dot{\mathcal{C}}^{\alpha}}= \sup_{\substack{x,y\in X\\ |x-y| \lt 1}} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}} = \inf\Big\{C\gt 0 \colon |f(x)-f(y)|\leq C |x-y|^\alpha\; \forall x,y\in X\Big\} \end{equation*}

and use it freely in the subsequent exercises.

3.2

Show that the space \(\mathcal{C}^{\alpha}([0,1];\R) \) with \(\alpha\gt 1\) consists only of constant functions.

Hint: Divide and conquer for any \(x,y\in[0,1]\) split the interval \([x,y]\) into \(2^M\) equal parts. Estimate \(|f(y)-f(x)|\) by comparing values at subsequent points in the splitting.

3.3 ​   long harder

Show that the space \(\Big(\mathcal{C}^{\alpha}([0,1];\R);\|\cdot\|_{\mathcal{C}^{\alpha}}\Big) \) is a Banach space.

Hints:

  • Write a line for each property of why \(\|\cdot\|_{\mathcal{C}^{\alpha}}\) is a norm (scaling, separation, triangle)
  • Show that a Cauchy sequence in \(\|\cdot\|_{\mathcal{C}^{\alpha}}\) is Cauchy in \(\|\cdot\|_{\sup}\).
  • Show that the limit function is in \(\mathcal{C}^{\alpha}\).
  • Show that the sequence converges in \(\|\cdot\|_{\mathcal{C}^{\alpha}}\) by passing to the limit.

3.4 ​   oneliner

Show that closed balls in \(\|\cdot\|_{\mathcal{C}^{\alpha}}\) are compact subsets of \(\Big(C^0([0,1];\R), \|\cdot\|_{\sup}\Big)\).

Hints:

  • Invoke Ascoli-Arzelà

3.5 ​   optional

Show that closed balls in \(\mathcal{C}^{\beta}([0,1];\R)\) are compact subsets of \(\mathcal{C}^{\alpha}([0,1];\R)\) with \(\alpha\lt \beta\).