Metric Spaces: Three Fundamental Theorems and Applications

Let $E$ be a compact subspace of a metric space $Y$. Show that $E$ is a closed, bounded subset of $Y$.

Show that an equicontinuous sequence of real-valued functions on a compact metric space is pointwise bounded if and only if it is uniformly bounded.

Show that an equicontinuous collection of continuous functions on a compact metric space is uniformly equicontinuous.

Let $X$ be a metric space and $\left\{ f_{n}\right\}$ a sequence in $C(X)$ that converges uniformly on $X$ to $f \in C(X)$. Show that $\left\{ f_{n}\right\}$ is equicontinuous.

A real-valued function $f$ on $[0,1]$ is said to be Hölder continuous of order $\alpha$ provided that there is a constant $C$ for which

Define the Hölder norm

Show that for $0 < \alpha \leq 1$, the set of functions for which $\left\| f\right\|_{\alpha } \leq 1$ has compact closure as a subset of $C[0,1]$.

Let $X$ be a compact metric space and $\mathcal{F}$ a subset of $C(X)$. Show that $\mathcal{F}$ is equicontinuous if and only if its closure in $C(X), \overline{\mathcal{F}}$, is equicontinuous. Conclude that a subset of $C(X)$ has compact closure if and only if it is equicontinuous and uniformly bounded.

For a closed, bounded interval $[a, b]$, let $\left\{ f_{n}\right\}$ be a sequence in $C[a, b]$. If $\left\{ f_{n}\right\}$ is equicontinuous, does $\left\{ f_{n}\right\}$ necessarily have a uniformly convergent subsequence? If $\left\{ f_{n}\right\}$ is uniformly bounded, does $\left\{ f_{n}\right\}$ necessarily have a uniformly convergent subsequence?

Let $X$ be a compact metric space and $Y$ be a general metric space. Denote by $C(X, Y)$ the set of continuous mappings from $X$ to $Y$. State and prove a version of the Arzelà–Ascoli Theorem for a sequence in $C(X, Y)$ in which the assumption that $\left\{ f_{n}\right\}$ is pointwise bounded is replaced by the assumption that for each $x \in X$, the closure of the set $\left\{ f_{n}(x) \mid n \text{ a natural number}\right\}$ is a compact subspace of $Y$.

Let $\left\{ f_{n}\right\}$ be an equicontinuous, uniformly bounded sequence of continuous real-valued functions on $\mathbb {R}$. Show that there is a subsequence of $\left\{ f_{n}\right\}$ that converges pointwise on $\mathbb {R}$ to a continuous function on $\mathbb {R}$ and that the convergence is uniform on each bounded subset of $\mathbb {R}$.

For $1 \leq p < \infty$, show that a subspace of $\ell^{p}$ is compact if and only if it is closed, bounded, and equisummable.

For a sequence of non-negative real numbers $\left\{ c_{n}\right\}$, let $\mathcal{S}$ be the subset of $\ell^{2}$ consisting of those $x = \left\{ x_{n}\right\} \in \ell^{2}$ such that $\left|x_{n}\right| \leq c_{n}$ for all $n$. Show that $\mathcal{S}$ is equisummable if $\left\{ c_{n}\right\}$ belongs to $\ell^{2}$.

For $1 \leq p \leq \infty$, show that the closed unit ball in $\ell^{p}$ is not compact.

For $1 \leq p \leq \infty$, show that the closed unit ball in the space $L^{p}[0,1]$ is not compact.

Let $S$ be a countable set and $\left\{ f_{n}\right\}$ a sequence of real-valued functions on $S$ that is pointwise bounded on $S$. Show that there is a subsequence of $\left\{ f_{n}\right\}$ that converges pointwise on $S$ to a real-valued function.

Let $p$ be a polynomial. Show that $p: \mathbb {R} \rightarrow \mathbb {R}$ is Lipschitz if and only if the degree of $p$ is less than 2.

Fix $\alpha > 0$, define $f(x) = \alpha x(1-x)$ for $x$ in $[0,1]$.

Does a mapping of a metric space $X$ into itself that is Lipschitz with Lipschitz constant less than 1 necessarily have a fixed-point?

Does a mapping of a complete metric space into itself that is Lipschitz with Lipschitz constant 1 necessarily have a fixed-point?

Let $X$ be a compact metric space and $T$ a mapping from $X$ into itself such that

Show that $T$ has a unique fixed-point.

Define $f(x) = \pi / 2+x-\arctan x$ for all real numbers $x$. Show that

Show that $f$ does not have a fixed-point. Does this contradict the preceding problem?

In Euclidean space $\mathbb {R}^{n}$ consider the closed unit ball $B = \left\{ x \in \mathbb {R}^{n} \mid \left\| x\right\| \leq 1\right\}$. Let $f$ map $B$ into $B$ and be Lipschitz with Lipschitz constant 1. Without using the Brouwer Fixed-Point Theorem, show that $f$ has a fixed-point.

Suppose that the mapping $f: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is a contraction. Define $g(x) = x-f(x)$ for all $x$ in $\mathbb {R}^{n}$. Show that the mapping $g: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is both one-to-one and onto. Also show that $g$ and its inverse are continuous.

Let $X$ be a complete metric space containing the point $x_{0}$ and let $r$ be a positive real number. Define $K = \left\{ x \text{ in } X \mid \rho \left(x, x_{0}\right) \leq r\right\}$. Suppose that the mapping $T: K \rightarrow X$ is Lipschitz with Lipschitz constant $c$. Suppose also that $c r+\rho \left(T\left(x_{0}\right), x_{0}\right) \leq r$. Prove that $T(K) \subseteq K$ and that $T: K \rightarrow X$ has a fixed-point.

Show that if the function $g: \mathbb {R}^{2} \rightarrow \mathbb {R}$ has continuous first-order partial derivatives, then for each point $\left(x_{0}, y_{0}\right)$ in $\mathbb {R}^{2}$ there is a neighborhood $\mathcal{O}$ of ($x_{0}, y_{0}$) on which the Lipschitz assumption (7) holds.

In case the function $g: \mathcal{O} \rightarrow \mathbb {R}$ has the form $g(x, y) = h(x)+b y$, where the function $h: \mathbb {R} \rightarrow \mathbb {R}$ is continuous, prove that the following is an explicit formula for the solution of (5):

Consider the differential equation

Show that the function $f: \mathbb {R} \rightarrow \mathbb {R}$ that is identically 0 is a solution and the function $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f(x) = 0$, if $x < 0$ and $f(x) = x^{3}$, if $x \geq 0$, is also a solution. Does this contradict the Picard Existence Theorem?

For a positive number $\epsilon$, consider the differential equation

Show that on the interval $I = (-\epsilon (\pi / 2), \epsilon (\pi / 2))$ there is a unique solution of this differential equation that is defined by $f(x) = \tan (x / \epsilon )$ and there is no solution in an interval strictly containing $I$.

Let $I$ be an open interval in $\mathbb {R}$ and suppose that the function $h: I \rightarrow \mathbb {R}$ has the property that there are points $x_{1} < x_{2}$ in $I$ and a number $c$ such that $h\left(x_{1}\right) < c < h\left(x_{2}\right)$ but $c$ does not belong to $h(I)$. Prove that there is no solution to the differential equation (5) by arguing that if $f: I \rightarrow \mathbb {R}$ is a solution, then the continuous function $f(x)-c x$ fails to attain a minimum value on the interval $\left[x_{1}, x_{2}\right]$.

Use the preceding exercise to prove that if $I$ is an open interval in $\mathbb {R}$ and the function $f: I \rightarrow \mathbb {R}$ is differentiable, then the image of the derivative $f^{\prime }: I \rightarrow \mathbb {R}$ is an interval.

State and prove a form of the Picard Existence Theorem for systems of differential equations in the following context: $\mathcal{O}$ is an open subset of $\mathbb {R} \times \mathbb {R}^{n}, \mathbf{g}: \mathcal{O} \rightarrow \mathbb {R}^{n}$ is continuous, the point ($x_{0}, \mathbf{y}_{0}$) is in $\mathcal{O}$, and the system of differential equations is

Let $E$ be a subset of a metric space $X$. Show that bd $E$ is closed. Also show that if $E$ is closed, then the interior of $\operatorname {bd} E$ is empty.

In a metric space $X$, show that a subset $E$ is nowhere dense if and only if for each open subset $\mathcal{O}$ of $X, E \cap \mathcal{O}$ is not dense in $\mathcal{O}$.

In a complete metric space $X$, is the union of a countable collection of nowhere dense sets also nowhere dense?

Let $\mathcal{O}$ be an open subset and $F$ be a closed subset of a metric space $X$. Show that both $\overline{\mathcal{O}} \sim \mathcal{O}$ and $F \sim \operatorname {int} F$ are closed and hollow.

In a complete metric space, is the union of a countable collection of sets of the first category also of the first category?

Let $F_{n}$ be the subset of $C[0,1]$ consisting of functions for which there is a point $x_{0}$ in $[0,1]$ such that $\left|f(x)-f(x_{0})\right| \leq n\left|x-x_{0}\right|$ for all $x \in [0,1]$. Show that $F_{n}$ is closed. Show that $F_{n}$ is hollow by observing that for $f \in C[0,1]$ and $r > 0$, there is a piecewise linear function $g \in C[0,1]$ for which $\rho_{\max }(f, g) < r$ and the left-hand and right-hand derivatives of $g$ on $[0,1]$ are greater than $n+1$. Conclude that $C[0,1] \neq \bigcup_{n = 1}^{\infty } F_{n}$ and show that each $h \in C[0,1] \sim \bigcup_{n = 1}^{\infty } F_{n}$ fails to be differentiable at any point in (0,1).

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. Conclude that there is not a real-valued function on $\mathbb {R}$ that is continuous just at the rational numbers.

For each $n$, show that in $[0,1]$ there is a nowhere dense closed set that has Lebesgue measure $1-1 / n$. Use this to construct a set of the first category in $[0,1]$ that has measure 1.

A point $x$ in a metric space $X$ is called isolated provided that the singleton set $\left\{ x\right\}$ is open in $X$.

Let $E$ be a subset of a complete metric space $X$. Verify the following assertions.

Show that under the hypotheses of Theorem 6 there is a dense open set $\mathcal{O} \subseteq X$ such that each $x \in \mathcal{O}$ has a neighborhood $U$ on which $\mathcal{F}$ is uniformly bounded.

Theorem 6 (A Non-linear Uniform Boundedness Theorem): Let $\mathcal{F}$ be a collection of continuous real-valued functions on a complete metric space $X$ that is pointwise bounded. Then there is a ball in $X$ on which $\mathcal{F}$ is uniformly pointwise bounded.

By Hölder's Inequality, $L^{2}[a, b] \subseteq L^{1}[a, b]$. Show that the set $L^{2}[a, b]$, considered as a subset of the complete metric space $L^{1}[a, b]$, is of the first category.

Let $f$ be a continuous real-valued function on $\mathbb {R}$ with the property that for each real number $x, \lim_{n \rightarrow \infty } f(n x) = 0$. Show that $\lim_{x \rightarrow \infty } f(x) = 0$.

Let $f$ be a continuous real-valued function on $\mathbb {R}$ that has derivatives of all orders. Suppose that for each real number $x$, there is an index $n = n(x)$ for which $f^{(n)}(x) = 0$. Show that $f$ is a polynomial.

For measurable sets $A$ and $B$ in a finite measure space ($X, \mathcal{M}, \mu$), show that

Let $\left\{ A_{n}\right\}$ be a sequence of measurable sets that converges to the measurable set $A_{0}$ with respect to the Nikodym metric. Show that $A_{0} \simeq \cup_{n = 1}^{\infty }\left[\cap_{k = n}^{\infty } A_{k}\right]$.

Let ($X, \mathcal{M}, \mu$) be a finite measure space and $\nu : \mathcal{M} \rightarrow [0, \infty )$ a finitely additive set-function with the property that for each $\epsilon > 0$, there is a $\delta > 0$ such that for a measurable set $E$, if $\mu (E) < \delta$, then $\nu (E) < \epsilon$. Show that $\nu$ is a measure on $\mathcal{M}$.

Let ($X, \mathcal{M}, \mu$) be a finite measure space. Show that the normed linear space $L^{1}(X, \mu )$ is separable if and only if the associated Nikodym metric space is separable.

Give an example of a decreasing sequence $\left\{ \mu_{n}\right\}$ of measures on $\mathcal{M}$ such that the set-function $\mu$ defined by $\mu (E) = \lim \mu_{n}(E)$ is not a measure.

Let $\left\{ \mu_{n}\right\}$ a sequence of measures on $\mathcal{M}$ such that for each $E \in \mathcal{M}, \mu_{n+1}(E) \geq \mu_{n}(E)$. For each $E \in \mathcal{M}$, define $\mu (E) = \lim \mu_{n}(E)$. Show that $\mu$ is a measure on $\mathcal{M}$ if $\mu (X) < \infty$.

Verify the assertion, in the proof of the Dunford-Pettis Theorem, that it suffices to consider non-negative functions.

Find a bounded sequence in $L^{1}([a, b], m)$, where $m$ is Lebesgue measure, which fails to have a weakly convergent subsequence.

Find a measure space ($X, \mathcal{M}, \mu$) for which every bounded sequence in $L^{1}(X, \mu )$ has a weakly convergent subsequence.

Let $(X, \mathcal{M}, \mu )$ be a finite measure space and $\left\{ \nu_{n}\right\}$ a sequence of finite measures of $\mathcal{M}$, each of which is absolutely continuous with respect to $\mu$. Prove that the following are equivalent: