Sequences and Series of Functions

Prove that every uniformly convergent sequence of bounded functions is uniformly bounded.

If $\left\{ f_{n}\right\}$ and $\left\{ g_{n}\right\}$ converge uniformly on a set $E$, prove that $\left\{ f_{n}+g_{n}\right\}$ converges uniformly on $E$. If, in addition, $\left\{ f_{n}\right\}$ and $\left\{ g_{n}\right\}$ are sequences of bounded functions, prove that $\left\{ f_{n} g_{n}\right\}$ converges uniformly on $E$.

Construct sequences $\left\{ f_{n}\right\} ,\left\{ g_{n}\right\}$ which converge uniformly on some set $E$, but such that $\left\{ f_{n} g_{n}\right\}$ does not converge uniformly on $E$ (of course, $\left\{ f_{n} g_{n}\right\}$ must converge on E).

Consider

For what values of $x$ does the series converge absolutely? On what intervals does it converge uniformly? On what intervals does it fail to converge uniformly? Is $f$ continuous wherever the series converges? Is $f$ bounded?

Let

Show that $\left\{ f_{n}\right\}$ converges to a continuous function, but not uniformly. Use the series $\Sigma f_{n}$ to show that absolute convergence, even for all $x$, does not imply uniform convergence.

Prove that the series

converges uniformly in every bounded interval, but does not converge absolutely for any value of $x$.

For $n = 1,2,3, \ldots , x$ real, put

Show that $\left\{ f_{n}\right\}$ converges uniformly to a function $f$, and that the equation

is correct if $x \neq 0$, but false if $x = 0$.

If

if $\left\{ x_{n}\right\}$ is a sequence of distinct points of $(a, b)$, and if $\Sigma \left|c_{n}\right|$ converges, prove that the series

converges uniformly, and that $f$ is continuous for every $x \neq x_{n}$.

Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions which converges uniformly to a function $f$ on a set $E$. Prove that

for every sequence of points $x_{n} \in E$ such that $x_{n} \rightarrow x$, and $x \in E$. Is the converse of this true?

Letting $(x)$ denote the fractional part of the real number $x$ (see Exercise 16, Chap.4, for the definition), consider the function

Find all discontinuities of $f$, and show that they form a countable dense set. Show that $f$ is nevertheless Riemann-integrable on every bounded interval.

Suppose $\left\{ f_{n}\right\} ,\left\{ g_{n}\right\}$ are defined on $E$, and

Suppose $g$ and $f_{n}(n = 1,2,3, \ldots )$ are defined on $(0, \infty )$, are Riemann-integrable on [ $t, T$ ] whenever $0 < t < T < \infty ,\left|f_{n}\right| \leq g, f_{n} \rightarrow f$ uniformly on every compact subset of $(0, \infty )$, and

Prove that

(See Exercises 7 and 8 of Chap. 6 for the relevant definitions). This is a rather weak form of Lebesgue's dominated convergence theorem (Theorem 11.32). Even in the context of the Riemann integral, uniform convergence can be replaced by pointwise convergence if it is assumed that $f \in \mathscr {R}$. (See the articles by F. Cunningham in Math. Mag., vol. 40, 1967, pp. 179-186, and by H. Kestelman in Amer. Math. Monthly, vol. 77, 1970, pp. 182-187).

Assume that $\left\{ f_{n}\right\}$ is a sequence of monotonically increasing functions on $\mathbb {R}^{1}$ with $0 \leq f_{n}(x) \leq 1$ for all $x$ and all $n$.

Let $f$ be a continuous real function on $\mathbb {R}^{1}$ with the following properties: $0 \leq f(t) \leq 1, f(t+2) = f(t)$ for every $t$, and

Put $\Phi (t) = (x(t), y(t))$, where

Prove that $\Phi$ is continuous and that $\Phi$ maps $I = [0,1]$ onto the unit square $I^{2} \subset \mathbb {R}^{2}$. If fact, show that $\Phi$ maps the Cantor set onto $I^{2}$.

Hint: Each $\left(x_{0}, y_{0}\right) \in I^{2}$ has the form

where each $a_{i}$ is 0 or 1. If

show that $f\left(3^{k} t_{0}\right) = a_{k}$, and hence that $x\left(t_{0}\right) = x_{0}, y\left(t_{0}\right) = y_{0}$. (This simple example of a so-called "space-filling curve" is due to I. J. Schoenberg, Bull. A.M.S., vol. 44, 1938, pp. 519).

Suppose $f$ is a real continuous function on $\mathbb {R}^{1}, f_{n}(t) = f(n t)$ for $n = 1,2,3, \ldots$, and $\left\{ f_{n}\right\}$ is equicontinuous on $[0,1]$. What conclusion can you draw about $f$?

Suppose $\left\{ f_{n}\right\}$ is an equicontinuous sequence of functions on a compact set $K$, and $\left\{ f_{n}\right\}$ converges pointwise on $K$. Prove that $\left\{ f_{n}\right\}$ converges uniformly on $K$.

Define the notions of uniform convergence and equicontinuity for mappings into any metric space. Show that Theorems 7.9 and 7.12 are valid for mappings into any metric space, that Theorems 7.8 and 7.11 are valid for mappings into any complete metric space, and that Theorems 7.10, 7.16, 7.17, 7.24, and 7.25 hold for vector-valued functions, that is, for mappings into any $\mathbb {R}^{k}$.

The cited results, as stated in the book for (real or complex) functions.

Theorem 7.8: The sequence of functions $\left\{ f_{n}\right\}$, defined on $E$, converges uniformly on $E$ if and only if for every $\varepsilon > 0$ there exists an integer $N$ such that $m \geq N$, $n \geq N$, $x \in E$ implies $\left|f_{n}(x)-f_{m}(x)\right| \leq \varepsilon$.

Theorem 7.9: Suppose $\lim_{n \rightarrow \infty } f_{n}(x) = f(x)$ $(x \in E)$. Put $M_{n} = \sup_{x \in E} \left|f_{n}(x)-f(x)\right|$. Then $f_{n} \rightarrow f$ uniformly on $E$ if and only if $M_{n} \rightarrow 0$ as $n \rightarrow \infty$.

Theorem 7.10: Suppose $\left\{ f_{n}\right\}$ is a sequence of functions defined on $E$, and suppose $\left|f_{n}(x)\right| \leq M_{n}$ $(x \in E, n = 1,2,3, \ldots )$. Then $\Sigma f_{n}$ converges uniformly on $E$ if $\Sigma M_{n}$ converges.

Theorem 7.11: Suppose $f_{n} \rightarrow f$ uniformly on a set $E$ in a metric space. Let $x$ be a limit point of $E$, and suppose that $\lim_{t \rightarrow x} f_{n}(t) = A_{n}$ $(n = 1,2,3, \ldots )$. Then $\left\{ A_{n}\right\}$ converges, and $\lim_{t \rightarrow x} f(t) = \lim_{n \rightarrow \infty } A_{n}$.

Theorem 7.12: If $\left\{ f_{n}\right\}$ is a sequence of continuous functions on $E$, and if $f_{n} \rightarrow f$ uniformly on $E$, then $f$ is continuous on $E$.

Theorem 7.16: Let $\alpha$ be monotonically increasing on $[a, b]$. Suppose $f_{n} \in \mathscr {R}(\alpha )$ on $[a, b]$, for $n = 1,2,3, \ldots$, and suppose $f_{n} \rightarrow f$ uniformly on $[a, b]$. Then $f \in \mathscr {R}(\alpha )$ on $[a, b]$, and $\int_{a}^{b} f d \alpha = \lim_{n \rightarrow \infty } \int_{a}^{b} f_{n} d \alpha$. (The existence of the limit is part of the conclusion.)

Theorem 7.17: Suppose $\left\{ f_{n}\right\}$ is a sequence of functions, differentiable on $[a, b]$ and such that $\left\{ f_{n}\left(x_{0}\right)\right\}$ converges for some point $x_{0}$ on $[a, b]$. If $\left\{ f_{n}^{\prime }\right\}$ converges uniformly on $[a, b]$, then $\left\{ f_{n}\right\}$ converges uniformly on $[a, b]$, to a function $f$, and $f^{\prime }(x) = \lim_{n \rightarrow \infty } f_{n}^{\prime }(x)$ $(a \leq x \leq b)$.

Theorem 7.24: If $K$ is a compact metric space, if $f_{n} \in \mathscr {C}(K)$ for $n = 1,2,3, \ldots$, and if $\left\{ f_{n}\right\}$ converges uniformly on $K$, then $\left\{ f_{n}\right\}$ is equicontinuous on $K$.

Theorem 7.25: If $K$ is compact, if $f_{n} \in \mathscr {C}(K)$ for $n = 1,2,3, \ldots$, and if $\left\{ f_{n}\right\}$ is pointwise bounded and equicontinuous on $K$, then (a) $\left\{ f_{n}\right\}$ is uniformly bounded on $K$, (b) $\left\{ f_{n}\right\}$ contains a uniformly convergent subsequence.

(Here $\mathscr {R}(\alpha )$ is the class of functions Riemann--Stieltjes integrable with respect to $\alpha$, and $\mathscr {C}(K)$ is the set of all complex-valued, continuous, bounded functions on $K$.)

Let $\left\{ f_{i n}\right\}$ be a uniformly bounded sequence of functions which are Riemann-integrable on $[a, b]$, and put

Prove that there exists a subsequence $\left\{ F_{n_{k}}\right\}$ which converges uniformly on [a, $b$ ].

Let $K$ be a compact metric space, let $S$ be a subset of $\mathscr {C}(K)$. Prove that $S$ is compact (with respect to the metric defined in Section 7.14) if and only if $S$ is uniformly closed, pointwise bounded, and equicontinuous. (If $S$ is not equicontinuous, then $S$ contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on $K$).

If $f$ is continuous on $[0,1]$ and if

prove that $f(x) = 0$ on $[0,1]$. Hint: The integral of the product of $f$ with any polynomial is zero. Use the Weierstrass theorem to show that $\int_{0}^{1} f^{2}(x) d x = 0$.

Let $K$ be the unit circle in the complex plane (i.e., the set of all $z$ with $\left|z\right| = 1$), and let $\mathscr {A}$ be the algebra of all functions of the form

Then $\mathscr {A}$ separates points on $K$ and $\mathscr {A}$ vanishes at no point of $K$, but nevertheless there are continuous functions on $K$ which are not in the uniform closure of $\mathscr {A}$. Hint: For every $f \in \mathscr {A}$

and this is also true for every $f$ in the closure of $\mathscr {A}$.

Assume $f \in \mathscr {R}(\alpha )$ on $[a, b]$, and prove that there are polynomials $P_{n}$ such that

(Compare with Exercise 12, Chap. 6).

Put $P_{0} = 0$, and define, for $n = 0,1,2, \ldots$,

Prove that

uniformly on $[-1,1]$. (This makes it possible to prove the Stone-Weierstrass theorem without first proving Theorem 7.26).

Hint: Use the identity

to prove that $0 \leq P_{n}(x) \leq P_{n+1}(x) \leq \left|x\right|$ if $\left|x\right| \leq 1$, and that

if $\left|x\right| \leq 1$.

Let $X$ be a metric space, with metric $d$. Fix a point $a \in X$. Assign to each $p \in X$ the function $f_{p}$ defined by

Prove that $\left|f_{p}(x)\right| \leq d(a, p)$ for all $x \in X$, and that therefore $f_{p} \in \mathscr {C}(X)$. Prove that

for all $p, q \in X$. If $\Phi (p) = f_{p}$ it follows that $\Phi$ is an isometry (a distance-preserving mapping) of $X$ onto $\Phi (X) \subset \mathscr {C}(X)$.

Let $Y$ be the closure of $\Phi (X)$ in $\mathscr {C}(X)$. Show that $Y$ is complete. Conclusion: $X$ is isometric to a dense subset of a complete metric space $Y$. (Exercise 24, Chap. 3 contains a different proof of this).

Suppose $\phi$ is a continuous bounded real function in the strip defined by $0 \leq x \leq 1,-\infty < y < \infty$. Prove that the initial-value problem

has a solution. (Note that the hypotheses of this existence theorem are less stringent than those of the corresponding uniqueness theorem; see Exercise 27, Chap. 5).

Hint: Fix $n$. For $i = 0, \ldots , n$, put $x_{i} = i / n$. Let $f_{n}$ be a continuous function on $[0,1]$ such that $f_{n}(0) = c$,

and put

except at the points $x_{i}$, where $\Delta_{n}(t) = 0$. Then

Choose $M < \infty$ so that $\left|\phi \right| \leq M$. Verify the following assertions.

Prove an analogous existence theorem for the initial-value problem

where now $c \in \mathbb {R}^{k}, y \in \mathbb {R}^{k}$, and $\Phi$ is a continuous bounded mapping of the part of $R^{k+1}$ defined by $0 \leq x \leq 1, y \in \mathbb {R}^{k}$ into $\mathbb {R}^{k}$. (Compare Exercise 28, Chap. 5). Hint: Use the vector-valued version of Theorem 7.25.