Chapter 7

For more exercises related to Picard's theorem see \(\$ 6.3\). Suppose J is a closed and bounded interval, and let \(Y:=\\{f \in C([-h, h], \mathbb{R}): f([-h, h]) \subset J\\} .\) Show that \(Y \subset C([-h, h], \mathbb{R})\) is closed. Hint: \(J\) is closed.

Let \((X, d)\) be a metric space and A a finite subset of \(X .\) Show that \(A\) is compact.

Show that for any set \(X,\) the discrete metric \((d(x, y)=1\) if \(x \neq y\) and \(d(x, x)=0)\) does give a metric space \((X, d)\).

Let \(A=\\{1 / n: n \in \mathbb{N}\\} \subset \mathbb{R}\). a) Show that \(A\) is not compact directly using the definition. b) Show that \(A \cup\\{0\\}\) is compact directly using the definition.

a) Show that \(d(x, y):=\min \\{1,|x-y|\\}\) defines a metric on \(\mathbb{R}\). b) Show that a sequence converges in \((\mathbb{R}, d)\) if and only if it converges in the standard metric. c) Find a bounded sequence in \((\mathbb{R}, d)\) that contains no convergent subsequence.

Let \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) be defined by \(f(0,0):=0,\) and \(f(x, y):=\frac{x y}{x^{2}+y^{2}}\) if \((x, y) \neq(0,0)\). a) Show that for any fixed \(x\), the function that takes y to \(f(x, y)\) is continuous. Similarly for any fixed \(y\), the function that takes \(x\) to \(f(x, y)\) is continuous. b) Show that \(f\) is not continuous.

Let \(X:=\\{0\\}\) be a set. Can you make it into a metric space?

Prove that in the proof of Picard's theorem, the statement"Without loss of generality assume \(x_{0}=0\) " is justified. That is, prove that if we know the theorem with \(x_{0}=0,\) the theorem is true as stated.

Let \((X, d)\) be a metric space with the discrete metric. a) Prove that \(X\) is complete. b) Prove that \(X\) is compact if and only if \(X\) is a finite set.

Suppose \(\left(X, d_{X}\right),\left(Y, d_{Y}\right)\) be metric spaces and \(f: X \rightarrow Y\) is continuous. Let \(A \subset X\). a) Show that \(f(\bar{A}) \subset \overline{f(A)}\) b) Show that the subset can be proper.

Let \(X:=\\{a, b\\}\) be a set. Can you make it into two distinct metric spaces? (define two distinct metrics on it)

Suppose \((X, d)\) is a nonempty metric space with the discrete topology. Show that \(X\) is connected if and only if it contains exactly one element.

Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(F(x):=k x+b\) where \(0

a) Show that the union of finitely many compact sets is a compact set. b) Find an example where the union of infinitely many compact sets is not compact.

Let the set \(X:=\\{A, B, C\\}\) represent 3 buildings on campus. Suppose we wish our distance to be the time it takes to walk from one building to the other. It takes 5 minutes either way between buildings \(A\) and \(B\). However, building \(C\) is on a hill and it takes 10 minutes from \(A\) and 15 minutes from \(B\) to get to \(C .\) On the other hand it takes 5 minutes to go from \(\mathrm{C}\) to \(\mathrm{A}\) and 7 minutes to go from \(\mathrm{C}\) to \(\mathrm{B}\), as we are going downhill. Do these distances define a metric? If so, prove it, if not, say why not.

Take \(\mathbb{Q}\) with the standard metric, \(d(x, y)=|x-y|,\) as our metric space. Prove that \(\mathbb{Q}\) is totally disconnected, that is, show that for every \(x, y \in \mathbb{Q}\) with \(x \neq y\), there exists an two open sets \(U\) and \(V\), such that \(x \in U, y \in V, U \cap V=\emptyset,\) and \(U \cap V=\mathbb{Q}\).

Let \(f:[0,1 / 4] \rightarrow[0,1 / 4]\) be defined by \(f(x):=x^{2} .\) a) Show that \(f\) is a contraction, and find the best (smallest) \(k\) from the definition that works. b) Find the fixed point and show directly that it is unique.

Suppose \(\left\\{x_{n}\right\\}_{n=1}^{\infty}\) converges to \(x .\) Suppose \(f: \mathbb{N} \rightarrow \mathbb{N}\) is a one-to-one function. Show that \(\left\\{x_{f(n)}\right\\}_{n=1}^{\infty}\) converges to \(x\).

Suppose \(f: X \rightarrow Y\) is continuous for metric spaces \(\left(X, d_{X}\right)\) and \(\left(Y, d_{Y}\right) .\) Show that if \(X\) is connected, then \(f(X)\) is connected.

Suppose \((X, d)\) is a metric space and \(\varphi:[0, \infty) \rightarrow \mathbb{R}\) is an increasing function such that \(\varphi(t) \geq 0\) for all \(t\) and \(\varphi(t)=0\) if and only if \(t=0 .\) Also suppose \(\varphi\) is subadditive, that is, \(\varphi(s+t) \leq\) \(\varphi(s)+\varphi(t) .\) Show that with \(d^{\prime}(x, y):=\varphi(d(x, y)),\) we obtain a new metric space \(\left(X, d^{\prime}\right)\).

a) Find an example of a contraction \(f: X \rightarrow X\) of a non-complete metric space \(X\) with no fixed point. b) Find a 1-Lipschitz map \(f: X \rightarrow X\) of a complete metric space \(X\) with no fixed point.

Show that a compact set \(K\) is a complete metric space (using the subspace metric).

If \((X, d)\) is a metric space where d is the discrete metric. Suppose \(\left\\{x_{n}\right\\}\) is a convergent sequence in \(X .\) Show that there exists \(a K \in \mathbb{N}\) such that for all \(n \geq K\) we have \(x_{n}=x_{K}\).

Let \(\left(X, d_{X}\right)\) and \(\left(Y, d_{Y}\right)\) be metric spaces. a) Show that \((X \times Y, d)\) with \(d\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right):=d_{X}\left(x_{1}, x_{2}\right)+d_{Y}\left(y_{1}, y_{2}\right)\) is a metric space. b) Show that \((X \times Y, d)\) with \(d\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right):=\max \left\\{d_{X}\left(x_{1}, x_{2}\right), d_{Y}\left(y_{1}, y_{2}\right)\right\\}\) is a metric space.

In any metric space, prove: a) \(E\) is closed if and only if \(\partial E \subset E\). b) \(U\) is open if and only if \(\partial U \cap U=\emptyset\).

Consider \(y^{\prime}=y^{2}, y(0)=1 .\) Use the iteration scheme from the proof of the contraction mapping principle. Start with \(f_{0}(x)=1 .\) Find a few iterates (at least up to \(f_{2}\) ). Prove that the pointwise limit of \(f_{n}\) is \(\frac{1}{1-x},\) that is for every \(x\) with \(|x|0,\) prove that \(\lim _{n \rightarrow \infty} f_{n}(x)=\frac{1}{1-x}\).

Let \(C([a, b], \mathbb{R})\) be the metric space as in Example 7.1.8. Show that \(C([a, b], \mathbb{R})\) is a complete metric space.

A set \(S \subset X\) is said to be dense in \(X\) if \(X \subset \bar{S}\) or in other words if for every \(x \in X,\) there exists a sequence \(\left\\{x_{n}\right\\}\) in \(S\) that converges to \(x .\) Prove that \(\mathbb{R}^{n}\) contains a countable dense subset.

A continuous function \(f: X \rightarrow Y\) for metric spaces \(\left(X, d_{X}\right)\) and \(\left(Y, d_{\gamma}\right)\) is said to be proper if for every compact set \(K \subset Y,\) the set \(f^{-1}(K)\) is compact. Suppose a continuous \(f:(0,1) \rightarrow(0,1)\) is proper and \(\left\\{x_{n}\right\\}\) is a sequence in (0,1) that converges to \(0 .\) Show that \(\left\\{f\left(x_{n}\right)\right\\}\) has no subsequence that converges in (0,1)

Let \(X\) be the set of continuous functions on \([0,1] .\) Let \(\varphi:[0,1] \rightarrow(0, \infty)\) be continuous. Define $$d(f, g):=\int_{0}^{1}|f(x)-g(x)| \varphi(x) d x$$ Show that \((X, d)\) is a metric space.

In any metric space, prove: a) Show that \(A\) is open if and only if \(A^{\circ}=A\). b) Suppose that \(U\) is an open set and \(U \subset A .\) Show that \(U \subset A^{\circ} .\)

Suppose \(f: X \rightarrow X\) is a contraction for \(k<1 .\) Suppose you use the iteration procedure with \(x_{n+1}:=f\left(x_{n}\right)\) as in the proof of the fixed point theorem. Suppose \(x\) is the fixed point of \(f .\) a) Show that \(d\left(x, x_{n}\right) \leq k^{n} d\left(x_{1}, x_{0}\right) \frac{1}{1-k}\) for all \(n \in \mathbb{N}\). b) Suppose \(d\left(y_{1}, y_{2}\right) \leq 16\) for all \(y_{1}, y_{2} \in X,\) and \(k=1 / 2 .\) Find an \(N\) such that starting at any point \(x_{0} \in X,\) \(d\left(x, x_{n}\right) \leq 2^{-16}\) for all \(n \geq N\)

Suppose \(\left\\{U_{n}\right\\}_{n=1}^{\infty}\) be a decreasing \(\left(U_{n+1} \subset U_{n}\right.\) for all \(\left.n\right)\) sequence of open sets in a metric space \((X, d)\) such that \(\bigcap_{n=1}^{\infty} U_{n}=\\{p\\}\) for some \(p \in X .\) Suppose \(\left\\{x_{n}\right\\}\) is a sequence of points in \(X\) such that \(x_{n} \in U_{n} .\) Does \(\left\\{x_{n}\right\\}\) necessarily converge to p? Prove or construct a counterexample.

Let \(\left(X, d_{X}\right)\) and \(\left(Y, d_{Y}\right)\) be metric space and \(f: X \rightarrow Y\) be a one-to-one and onto continuous function. Suppose \(X\) is compact. Prove that the inverse \(f^{-1}: Y \rightarrow X\) is contimuous.

Let \((X, d)\) be a metric space. For nonempty bounded subsets \(A\) and \(B\) let $$ d(x, B):=\inf \\{d(x, b): b \in B\\} \quad \text { and } \quad d(A, B):=\sup \\{d(a, B): a \in A\\}$$ Now define the Hausdorff metric as $$d_{H}(A, B):=\max \\{d(A, B), d(B, A)\\}$$ Note: \(d_{H}\) can be defined for arbitrary nonempty subsets if we allow the extended reals. a) Let \(Y \subset \mathscr{P}(X)\) be the set of bounded nonempty subsets. Prove that \(\left(Y, d_{H}\right)\) is a so-called pseudometric space: \(d_{H}\) satisfies the metric properties \((i),(i i i),(i v),\) and further \(d_{H}(A, A)=0\) for all \(A \in Y\). b) Show by example that d itself is not symmetric, that is \(d(A, B) \neq d(B, A) .\) c) Find a metric space \(X\) and two different nonempty bounded subsets \(A\) and \(B\) such that \(d_{H}(A, B)=0 .\)

Let \(X\) be a set and \(d, d^{\prime}\) be two metrics on \(X .\) Suppose there exists an \(\alpha>0\) and \(\beta>0\) such that \(\alpha d(x, y) \leq d^{\prime}(x, y) \leq \beta d(x, y)\) for all \(x, y \in X .\) Show that \(U\) is open in \((X, d)\) if and only if \(U\) is open in \(\left(X, d^{\prime}\right) .\) That is, the topologies of \((X, d)\) and \(\left(X, d^{\prime}\right)\) are the same.

Let \(f(x):=x-\frac{x^{2}-2}{2 x}\) (you may recognize Newton's method for \(\sqrt{2}\) ). a) Prove \(f([1, \infty)) \subset[1, \infty) .\) b) Prove that \(f:[1, \infty) \rightarrow[1, \infty)\) is a contraction. c) Apply the fixed point theorem to find an \(x \geq 1\) such that \(f(x)=x,\) and show that \(x=\sqrt{2}\).

Show that there exists a metric on \(\mathbb{R}\) that makes \(\mathbb{R}\) into a compact set.

Let \(E \subset X\) be closed and let \(\left\\{x_{n}\right\\}\) be a sequence in \(X\) converging to \(p \in X .\) Suppose \(x_{n} \in E\) for infinitely many \(n \in \mathbb{N}\). Show \(p \in E\).

Take the metric space of continuous functions \(C([0,1], \mathbb{R}) .\) Let \(k:[0,1] \times[0,1] \rightarrow \mathbb{R}\) be \(a\) contimuous function. Given \(f \in C([0,1], \mathbb{R})\) define $$ \varphi_{f}(x):=\int_{0}^{1} k(x, y) f(y) d y $$ a) Show that \(T(f):=\varphi_{f}\) defines a function \(T: C([0,1], \mathbb{R}) \rightarrow C([0,1], \mathbb{R})\). b) Show that \(T\) is continuous.

Let \((X, d)\) be a nonempty metric space and \(S \subset X\) a subset. Prove: a) \(S\) is bounded if and only if for every \(p \in X,\) there exists \(a B>0\) such that \(d(p, x) \leq B\) for all \(x \in S\). b) A nonempty \(S\) is bounded if and only if \(\operatorname{diam}(S):=\sup \\{d(x, y): x, y \in S\\}<\infty\).

Suppose \(\left\\{S_{i}\right\\}, i \in \mathbb{N}\) is a collection of connected subsets of a metric space \((X, d) .\) Suppose there exists an \(x \in X\) such that \(x \in S_{i}\) for all \(i \in \mathbb{N}\). Show that \(\bigcup_{i=1}^{\infty} S_{i}\) is connected.

Suppose \(f: X \rightarrow X\) is a contraction, and \((X, d)\) is a metric space with the discrete metric, that is \(d(x, y)=1\) whenever \(x \neq y .\) Show that \(f\) is constant, that is, there exists a \(c \in X\) such that \(f(x)=c\) for all \(x \in X\).

Suppose \((X, d)\) is complete and suppose we have a countably infinite collection of nonempty compact sets \(E_{1} \supset E_{2} \supset E_{3} \supset \cdots .\) Prove \(\bigcap_{j=1}^{\infty} E_{j} \neq \emptyset\)

Take \(\mathbb{R}^{*}=\\{-\infty\\} \cup \mathbb{R} \cup\\{\infty\\}\) be the extended reals. Define \(d(x, y):=\left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right|\) if \(x, y \in \mathbb{R},\) define \(d(\infty, x):=\left|1-\frac{x}{1+|x|}\right|, d(-\infty, x):=\left|1+\frac{x}{1+|x|}\right|\) for all \(x \in \mathbb{R},\) and let \(d(\infty,-\infty):=2\) a) Show that \(\left(\mathbb{R}^{*}, d\right)\) is a metric space. b) Suppose \(\left\\{x_{n}\right\\}\) is a sequence of real numbers such that for every \(M \in \mathbb{R},\) there exists an \(N\) such that \(x_{n} \geq M\) for all \(n \geq N .\) Show that \(\lim x_{n}=\infty\) in \(\left(\mathbb{R}^{*}, d\right)\) c) Show that a sequence of real numbers converges to a real number in \(\left(\mathbb{R}^{*}, d\right)\) if and only if it converges in \(\mathbb{R}\) with the standard metric.

a) Working in \(\mathbb{R}\), compute \(\operatorname{diam}([a, b])\). b) Working in \(\mathbb{R}^{n},\) for any \(r>0,\) let \(B_{r}:=\left\\{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}

Let \(A\) be a connected set in a metric space. a) Is \(\bar{A}\) connected? Prove or find a counterexample. b) Is A \(^{\circ}\) connected? Prove or find a counterexample. Hint: Think of sets in \(\mathbb{R}^{2}\).

Suppose \(\left\\{V_{n}\right\\}_{n=1}^{\infty}\) is a collection of open sets in \((X, d)\) such that \(V_{n+1} \supset V_{n} .\) Let \(\left\\{x_{n}\right\\}\) be a sequence such that \(x_{n} \in V_{n+1} \backslash V_{n}\) and suppose \(\left\\{x_{n}\right\\}\) converges to \(p \in X .\) Show that \(p \in \partial V\) where \(V=\bigcup_{n=1}^{\infty} V_{n}\).

Let \((X, d)\) be a compact metric space, let \(C(X, \mathbb{R})\) be the set of real-valued continuous functions. Define $$ d(f, g):=\|f-g\|_{u}:=\sup _{x \in X}|f(x)-g(x)| $$ a) Show that d makes \(C(X, \mathbb{R})\) into a metric space. b) Show that for any \(x \in X,\) the evaluation function \(E_{x}: C(X, \mathbb{R}) \rightarrow \mathbb{R}\) defined by \(E_{x}(f):=f(x)\) is \(a\) continuous function.

a) Find a metric d on \(\mathbb{N}\), such that \(\mathbb{N}\) is an unbounded set in \((\mathbb{N}, d)\). b) Find a metric \(d\) on \(\mathbb{N}\), such that \(\mathbb{N}\) is a bounded set in \((\mathbb{N}, d)\). c) Find a metric \(d\) on \(\mathbb{N}\) such that for any \(n \in \mathbb{N}\) and any \(\varepsilon>0\) there exists an \(m \in \mathbb{N}\) such that \(d(n, m)<\varepsilon\).