The Space $C(X)$

Prove that $\left\| f\right\| :=\max \left\{ \left|f(x)\right|: x \in X\right\}$ defines a norm on $C(X)$, if $X$ is a compact metric space.

Let $X$ be a compact metric space. Prove that $C(X)$ is complete in the topology induced by the norm.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=\frac{x^{2}}{x^{2}+(1-n x)^{2}}, X=[0,1]$.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=x^{2}+\frac{1}{n}, X=\mathbb {R}$.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=\arctan (n x), X=[0,1]$.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=\frac{\sin n x}{n^{2}}, X=\mathbb {R}$.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=n^{2} x\left(1-x^{2}\right)^{n}, X=[0,1]$.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=x+\frac{1}{n x}, X=[1,2]$.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=x^{n}, X=[0,1 / 2]$.

In Exercises 10.7.3-10.7.10 determine whether the family of functions $\mathcal{F}= \left\{ f_{n}: n \in \mathbb {N}\right\}$ is equicontinuous on $X$.

$f_{n}(x)=\frac{n x}{1+n^{3} x^{2}}, X=[0,1]$.

Determine whether the family of functions $\mathcal{F}=\left\{ \int_{0}^{x} f(t) d t: f \in C[0,1],\left\| f\right\|_{\text{max }} \leq 1\right\}$ is equicontinuous on [ 0,1 ].

Let $\mathcal{F}$ be an equicontinuous collection of functions in $C(X)$ and let $\overline{\mathcal{F}}$ denotes its closure. Prove that $\overline{\mathcal{F}}$ is equicontinuous.

Let $\mathcal{F}$ be a family of continuous functions on $[0,1]$ having the property that there exists $M>0$ such that $\left|f(x)\right| \leq M$ for all $f \in \mathcal{F}$ and all $x \in [0,1]$. Let

Let $\mathcal{F}$ be a collection of functions in $C^{1}[0,1]$, let $M>0$, and suppose that $\left|f^{\prime }(x)\right| \leq M$, for all $f \in \mathcal{F}$ and all $x \in [0,1]$. Prove that $\mathcal{F}$ is equicontinuous.

Let $\mathcal{F}$ be a collection of functions in $C^{1}[0,1]$, let $x_{0} \in [0,1]$, let $M>0$, and suppose that $\left|f^{\prime }\left(x_{0}\right)\right| \leq M$, for all $f \in \mathcal{F}$. Prove or disprove: $\mathcal{F}$ is equicontinuous at $x_{0}$.

A collection $\mathcal{F}$ of real-valued functions on a metric space $X$ is uniformly equicontinuous if

Prove that if $X$ is compact every equicontinuous family of continuous functions is uniformly equicontinuous.

Let $X$ be a compact metric space and $\mathcal{F}$ an equicontinuous sequence of real-valued functions. Prove that $\mathcal{F}$ is pointwise bounded if and only if it is uniformly bounded.

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

Let $\left\{ f_{n}\right\}$ be a sequence in $C[a, b]$. Must $\left\{ f_{n}\right\}$ have a uniformly convergent subsequence if:

For $1 \leq p<\infty$, a collection $\mathcal{S} \subset \ell^{p}$ is equisummable if

Prove that $\mathcal{S}$ is compact if and only if it is closed, bounded, and equisummable.

Let $\left\{ c_{n}\right\}$ be a sequence of nonnegative numbers in $\ell^{2}$ and let $\mathcal{S}$ be a subset of $\ell^{2}$ consisting of those $\left\{ x_{n}\right\} \in \ell^{2}$ satisfying $\left|x_{n}\right| \leq c_{n}$, for all $n \in \mathbb {N}$. Prove that $\mathcal{S}$ is equisummable.

Let $\left\{ f_{n}\right\} \subset C(X)$ and suppose that $\lim_{n} f_{n}(x)=f(x)$ for all $x \in X$. Prove that there is a dense subset $D$ of $X$ such that $\left\{ f_{n}\right\}$ is equicontinuous on $D$ and $f$ is continuous on $D$.

Prove Theorem 10.7.10.

Theorem 10.7.10 (Arzelà-Ascoli Theorem): Let $X$ be a compact metric space and $\mathcal{F} \subset C(X)$. Then $\mathcal{F}$ is compact if and only if it is closed, uniformly bounded, and equicontinuous.

Let $\left\{ a_{n}\right\}$ be a sequence of real numbers in $[1,2]$ and let the sequence $\left\{ f_{n}\right\} \subset C[0,1]$ be defined by

Prove that it has a convergent subsequence.

Let $K$ be a continuous function on $[0,1] \times [0,1]$, and let $\left\{ f_{n}\right\}$ be a bounded sequence in $C[0,1]$, and define

Prove that $\left\{ g_{n}\right\}$ is a sequence in $C[0,1]$ that has a convergent subsequence.

Let $f \in L^{1}\left(0,1\right)$, let $\left\{ E_{n}\right\}$ be measurable subsets of $[0,1]$, and let

Prove that $\left\{ g_{n}\right\}$ has a subsequence that converges uniformly to a continuous function on $[0,1]$.

Let $\left\{ f_{n}\right\}$ be an equicontinuous, uniformly bounded sequence of realvalued functions on $\mathbb {R}$. Prove that there is a subsequence $\left\{ f_{n_{k}}\right\}$ that converges pointwise on $\mathbb {R}$ to a continuous function $f$ on $\mathbb {R}$ and that the convergence is uniform on bounded subsets of $\mathbb {R}$.

Let $X$ be a compact metric space. Prove that $C(X)$ is separable.

Let $\mu$ be a signed Borel measure on $[0,1]$ such that

for all $x>1$. Show that $\mu =0$.

Let $g: \mathbb {R} \rightarrow [0,+\infty )$ and suppose that, for $f \in C^{1}[0,1]$,

Finally, let $\mathcal{F}=\left\{ f \in C^{1}[0,1]:\left\| f\right\| \leq 1\right\}$. Prove that there is a function $h \in \mathcal{F}$ that minimizes $I(f)$, i.e., $I(h)=\inf \left\{ I(f): f \in \mathcal{F}\right\}$.

Prove that the collection of finite sums of the form $\sum_{i=1}^{n} g_{i}(x) h_{i}(y)$, where $n \in \mathbb {N}, g_{i} \in C[0,1]$, and $h_{i} \in C[0,1]$, is dense in $C([0,1] \times [0,1])$.