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\input amstex.tex
\input amssym.tex


\baselineskip20pt
\tolerance=10000
\hbadness=10000
\vbadness=10000
\magnification=1200
\hsize = 5.25in
\hoffset = .5in
\def\NN{I\hskip-.13cmN}
\def\RR{I\hskip-.13cmR}
%\input pictex
\footline={}
\nopagenumbers
\def \pni{\par\noindent}\newline 
\centerline {\bf {Analysis Test Bank}}
\bigskip
\rightline{Preliminary Version, September 1999}
\vskip.5 true in

\item{1.}  Metric and normed spaces, completeness and completion
\item{2.}  Continuous linear transformations and functions, closed graph theorem, open mapping theorem
\item{3.}  Continuous functions: uniform convergence, principle of uniform boundedness, Stone-Weierstrass, compactness
\item{4.} Differentiable functions: Jacobians, inverse and implicit functions, 
power series
\item{5.}  Functions of bounded variation
\item{6.}  Riemann integral, Lebesgue integral (via completion), convergence theorems
\item{7.} Absolutely continuous functions and the fundamental theorem of calculus
\item{8.} Basic properties of $L^p$-spaces, Riesz representation for 
$L^p$-spaces
\bigskip
\bigskip
\noindent
{\sl Suggested Reading}
\bigskip\noindent
D.S. Bridges: Foundations of Real and Abstract Analysis. Springer, 1997

\noindent
A. Browder: Mathematical Analysis. Springer, 1996.


\noindent
I.P. Natanson: Theory of Functions of a Real Variable, Vol. 1, 1955.

\noindent
H.L. Royden: Real Analysis, Macmillan, 1988.

\noindent
W. Rudin: Principles of Mathematical Analysis. McGraw-Hill, 1953.

\noindent
G.F. Simmons: Introduction to Topology and Modern Analysis. McGraw-Hill, 1963.

\noindent
R.L. Wheeden and A. Zygmund: Measure and Integral: An Introduction to Real Analysis. Marcel Dekker, 1977.

\vfill
\eject
\headline={\hfill\tenrm{}\folio}
\pageno=1
\pni

\centerline {\bf {1.  Metric and normed spaces, completeness and completion}}
\pni\bigskip\bigskip

\item{1.1}  Let $X$ be a compact set and $C(X)$ the space of real continuous 
functions on $X$ equipped with the $\sup$ norm.  Let ${\cal{M}} = \{f \in 
C(X)$  such that $f (x) > 0$ for all $x \in X \}$. Show that $\cal{M}$ is an open subset of $C(X)$.




\bigskip

\item{1.2}  Let $X$ be the normed linear space obtained by putting the norm 
 $\| f \|_{0}:=
\sup_{t \in [0,1]} | \int_{0}^{t} f(s)\, ds|$ on the set of continuous real
functions on $[0,1]$.
\medskip
\itemitem{(a)} Show that the functions $f_n(t) := \sin (nt)$ converge to zero in $X$.
\itemitem{(b)}  Show that $X$ is not a Banach space.
\itemitem{(c)} Let $f: [0,1]\to\RR$ be continuous and extend $f$ to $[0,\infty)$ by setting $f(t) := f(1)$ for $t > 1$. Show that the differential quotients $D_h: t \to {{f(t+h) - f(t)}\over h}$ converge uniformly on [0,1] as $h \to 0^+$ if $f$ is continuously differentiable. Show that  the differential quotients $D_h$ are Cauchy in $X$ as $h \to 0^+$ for any continuous $f : [0,1]\to\RR$ with $f(0) = 0$; i.e., for all $\epsilon > 0$ there exists $h_0 > 0$ such that $\|D_h - D_k\|_{0} < \epsilon$ for all $0<h,k<h_0$.

\bigskip

\item{1.3}    For $0 < \alpha \leq 1$ consider the 
space $L_\alpha$ of all functions on $[0,1] \to \RR$ such that
$$\|f\|_\alpha := |f(0)| + \sup_{s \neq t} {{| f(s) - f(t) |} \over {| s - t |^\alpha}} < \infty.$$
\itemitem{(a)}  Show that $L_\alpha$ is a Banach 
space.
\itemitem{(b)}  Show that each element in $L_{\alpha}$ is absolutely continuous.
\vfill
\break


\pni
\centerline {\bf {2.  Continuous linear transformations and functions}}
\centerline{\bf{ closed graph theorem, open mapping theorem}}

\pni\bigskip\bigskip
\item{2.1} (a)  Let $X$ be a complete metric space. A mapping $F: X \rightarrow X$  is said to be a  
\itemitem{} contraction if there is a constant $r < 1$ such that $d(F(u), F(v)) \leq 
r \cdot d(u,v)$ for all $u, v \in X$.  Given a contraction $F$ and a point 
$u_ {0} \in X$, define a sequence $(u_k)_{k\in \NN}$ in $X$ by $u_{k + 1} = F(u_k)$.  Show that $d({u_{k+1},u_k}) \le r^kd(u_1,u_0)$ and
 prove that the sequence $(u_k)$ converges to the unique fixed point of $F$.
\itemitem{(b)}   Let $g\in C[0,1]$ with $\int_0^1 |g(s)|\, ds \le r < 1$. Use part (a) to show that, for all $f\in C[0,1]$, there exists a unique 
solution $u = u(\cdot) \in C [0,1]$ of the equation
$$u(t) = \int_{0}^{t} g(t - s) u (s) ds + f(t),~~~~0 \leq t \leq 1.      \eqno         (*)$$
\itemitem{(c)}  Show that the operator  $A$ which assigns to each $f \in C[0,1]$ 
the unique solution $u$ of the equation $(*)$ is a linear operator from $C[0,1]$ into 
$C [0,1 ]$.
\itemitem{(d)}  Use the `Closed Graph Theorem' to show that $A$ is a continuous linear operator.
\itemitem{(e)} Show that the solutions $u$ of $(*)$ depend continuously on the forcing terms $f$.

\bigskip

\item{2.2}  Let $k$ be a measurable function on $\RR^2$ such 
that $\int_{\RR} (\int_{\RR} | k(x,y) 
|^q dy)^{p \slash q} dx < \infty$ for some $1 < q < \infty$ and 
${1 \over p} + {1 \over  q} + 1$.  Show that
$$(T f)(x) := \int_{\RR} k(x,y) f(y) dy$$
\item{}  defines a continuous linear map $T : L_p (\RR) \rightarrow L_p 
(\RR)$. 

\bigskip

\item{2.3}  Let $X$ be the normed linear space obtained by putting the norm 
$\| f \|_1 = \int_{0}^{1} | f(t)|\, dt$ on the set of continuous real
functions on $[0,1]$.
\medskip
\itemitem{(a)}  Show that $X$ is not a Banach space. {\bf [$\to$ 1]}
\itemitem{(b)}  Show that the linear functional $\Lambda f = f (1 \slash 2)$ is 
not bounded.

\vfill
\eject
\centerline {\bf {3.  Continuous functions: uniform convergence, principle of uniform}}
\centerline {\bf {boundedness, Stone-Weierstrass, compactness}}
\pni\bigskip\bigskip
\item{3.1}  A subset $S$ of $\RR$ is of type $F_\sigma$ if $S$ is the 
countable union of closed sets.
\medskip
\itemitem{(a)}  Let $f$ be any function from $\RR$ to $\RR$.  Prove that the set of points of discontinuity of $f$ 
is of type $F_\sigma$.
\itemitem{(b)}  Can a function from $\RR$ to $\RR$ be 
continuous on the rationals and discontinuous on the irrationals?  What if the 
roles of the rationals and irrationals are interchanged?
\itemitem{(c)}  Briefly explain why there are continuous, nowhere differentiable 
functions on $\RR$.

\bigskip

\item{3.2}    
(a)  Let $f_n : \RR\to\RR$ be given by $f_n (x) = {x \over n}$ ($n\in\NN$).  
Show that the sequence 
\itemitem{} $(f_n)_{n\in\NN}$ is  pointwise convergent on $\RR$ but not uniformly convergent 
on $\RR$.
\itemitem{(b)}  Let $f_n : [0,1]\to\RR$ be given by $f_n (x) = {1 \over {1 + nx^2}}$ ($n\in\NN$).  Show that $(f_n)_{n\in\NN}$ is a bounded subset of $C [0,1]$ and that no subsequence 
of $(f_n)_{n\in\NN}$ converges in $C[0,1]$.

\bigskip

\item{3.3}  Let $f_n : \RR\to\RR$ be given by $f_n (x) = {x \over {1 + nx^2}}$ ($n\in\NN$).
\medskip
\itemitem{(a)}  Show that $\sup_{x\in\RR} | f_n(x) | 
 = {1 \over {2 \sqrt{n}}}$.  Conclude that the sequence $(f_n)_{n\in\NN}$ converges uniformly on 
$\RR$ to a function $f$.  What is $f$?
\itemitem{(b)}  Show that the equation $f^{\prime} (x) = \lim_{n~\rightarrow~\infty} 
f_n^{\prime} (x)$ is true if $x \neq 0$, but false if $x = 0$.

\bigskip

\item{3.4}  Identify all subsets of $[0,1]$ on which $\sum_{n = 0}^{\infty} x^n$ 
converges uniformly.  Explain.

\bigskip 

\item{3.5}  (a) 
  Show that every continuous function on $[0,1]$ is a uniform limit 
of step functions.
\itemitem{(b)}  Is the converse true?  (A step function is finite linear combination of 
characteristic functions of intervals.)

\bigskip

\item{3.6}  Show that every continuous function $f : [0,1] \times [0,1] 
\to {\RR}$ can be uniformly approximated by polynomials $p(s,t) = \alpha_{0} + \sum_{n,m = 1}^{N} \alpha_{n m} t^n x^m$, where 
$\alpha_{n m} \in {\RR}$ and $N \in {\NN}$.  Is the same result 
true for continuous functions $f : {\RR} \times {\RR} \rightarrow {\RR}$?


\bigskip

\item{3.7}  Prove or disprove: The product of two uniformly continuous functions 
on $\RR$ is also uniformly continuous.

\bigskip

\item{3.8} Let $ \chi_{[-n,n]} (\cdot)$ denote the characteristic function of the interval $[-n,n]$ ($n\in\NN$). Consider the sequence of functions
$f_{n}(x) := \chi_{[-n,n]} (x) \sin ({\pi x \over n})$
($x \in \RR$).

\itemitem{(a)}  Determine $f(x) = \lim_{n \rightarrow \infty} f_n(x)$ and  show that the sequence $(f_n)_{n\in\NN}$ converges uniformly on compact subsets of $\RR$.  Does the sequence converge uniformly on $\RR$?
\itemitem{(b)}  Show that
$$\int_{- \infty}^{\infty} f(x) \, dx = \lim_{n \rightarrow \infty} \int_{- \infty}^{\infty} 
f_n (x) \, dx.$$
\itemitem{}  Are the assumptions of Lebesgue's dominated convergence theorem 
satisfied?

\bigskip

\item{3.9}  Prove or disprove the following two statements:
\medskip
\itemitem{(a)}  If $\sum_{n = 1}^{\infty} a_n$ converges, then 
$\sum_{n = 1}^{\infty} a_n \cos n x$ converges pointwise everywhere on 
$\RR$.
\itemitem{(b)}  If $\sum_{n = 1}^{\infty} a_n$ converges absolutely, then 
$\sum_{n = 1}^{\infty} a_n \cos n x$ converges to a continuous function 
on $\RR$.

\bigskip

\item{3.10}  Let $f(x)=\left\{\aligned
x^2\text{\rm , if $x$ is irrational}\\
0\text{\rm , if x is rational  }\endaligned
\right.$
Show that $f$ is continuous at only one point, and that $f$ is 
differentiable there.

\bigskip

\item{3.11}   Let $C_{0,0}[0,1]$ be the space of all 
continuous real functions $f$ on the internal $[0,1]$ satisfying $f(0) = f (1) = 0$.  
Let ${\cal P}_{0,0}$ be the subspace of polynomials in $C_{0,0}[0,1]$.  Show that
${\cal P}_{0,0}$ is dense in $C_{0,0}[0,1]$ in the sup norm.

\bigskip

\item{3.12}  Let $f_n : [0,\infty) \to \RR$ be defined by $f_n(x) := (x \slash n)e^{-(x \slash n)}$ ($n\in\NN$).
\itemitem{(a)}  Determine $f(x) = \lim_{n \rightarrow \infty} f_n(x)$.  Show that the sequence 
$(f_n)_{n\in\NN}$ converges uniformly to $f$ on $[0,a]$ for any non-negative real number $a$.  
Does the sequence converge uniformly to $f$ on $[0, \infty)$?  Justify your answer.
\itemitem{(b)}  Show that $f(x) = \lim_{n \to \infty} \int_0^a f_n (x) \, dx  = 
\int_{0}^{a} f(x) \, dx$, but that $\lim_{n \to \infty} \int_{0}^{\infty}
 f_n(x) \, dx \neq \int_{0}^{\infty} f(x) \, dx$.

\bigskip

\item{3.13}  Let $I = [0,1]$.  Suppose $f$ is a continuous real-valued function 
on $I \times I$.  Show that $f$ can be uniformly approximated by functions of the 
from $\sum_{i = 1}^{n}~f_i(x) g_i (y)$ where $f_i$ and $g_i$ are continuous real-valued
function on $I$.


\bigskip

\item{3.14}  Show: if $f \in C [0,1]$ and $\int_{0}^{1} f(x) x^n dx = 0$ for all
$n\in \NN_0$, then $f = 0$.  


\bigskip


\item{3.15}  Let $(x_n)_{n\in\NN}$ be a sequence of real numbers.  Prove that 
the following are equivalent.
\medskip
\itemitem{(i)}  $\lim_{n\to\infty} x_n = a$.
\itemitem{(ii)}  Every subsequence of $(x_n)_{n\in\NN}$ contains a subsequence that 
converges to $a$.

\bigskip

\item{3.16}  Show that the function
$f(x) = 2^{-x} + 3 \cdot 2^{-3 x} + \ldots + (2n + 1)2^{-(2n + 1)x} + \ldots$
 is continuous on $(0, \infty)$.

\bigskip

\item{3.17}   (a)   
  Prove or disprove:  A continuous function on the 
interval $[0, \infty)$ can be 
\itemitem{} approximated uniformly by polynomials.
\itemitem{(b)}  Prove or disprove: If $f$ and $g$ are both functions from 
$\RR$ into $\RR$, and $\lim_{t \to a} g(t) = b$ and  $\lim_{t \to b} f(t) = c$, then $\lim_{t\to a}
f(g(t)) = c$.

\bigskip

\item{3.18}  Let $f : [0,1]\to \RR$ be continuously differentiable.  Prove that 
for any $\epsilon > 0$, there exists a polynomial $P(x)$ such that $\| f - P 
\|_{\infty} < \epsilon$ and $\| f^{\prime} - P^{\prime} \|_{\infty} 
< \epsilon $.  Here $\| \cdot \|_{\infty}$ 
denotes the sup-norm.

\bigskip

\item{3.19}  Prove: If $f \in C[0,1]$ and $\int_{0}^{1} f(x) e^{-nx} \,
dx = 0$ for all $n \in \NN_0$, then $f = 0$.

\bigskip

\item{3.20}  Let $f_n : [1,\infty) \to \RR$ be defined by $f_n (x) := {{n + 1} \over n} e^{-nx}$  ($n\in\NN$).  Show 
that the series $\sum_{k = 1}^{\infty} f_k$ converges uniformly to a continuous 
function.

\bigskip

\item{3.21}  $f_n : \RR \to \RR$ be defined by $f_n (x):= (\sin x)^n $  ($n\in\NN$).  Does $(f_n)_{n\in\NN}$ converge uniformly?


\vfil
\break


\centerline{\bf {4. Differentiable functions: Jacobians, inverse and implicit functions, 
power series}}
\pni\bigskip\bigskip
\item{4.1}  Does $e^z (x^2 + y^2 + z^2 ) - \sqrt{1 + z^2} + y = 0$ have 
a solution $z = f(x,y)$ which is continuous at $x = 1, y = 0$ and $f(1,0)=0$?  
Explain carefully!

\bigskip

\item{4.2}  Let $f : \RR \rightarrow \RR$ be an infinitely differentiable function.
\medskip
\itemitem{(a)}  Use Taylor's formula with remainder to show that given $x$ and $h$ 
then $f^{\prime} (x) = (f(x + 2h) - f(x)) \slash 2h - hf^{\prime \prime} (\xi)$ for some $\xi$.
\itemitem{(b)}  Assume $f(x) \rightarrow 0$ as $x \rightarrow \infty$, and that 
$f^{\prime \prime}$ is bounded.  Show that $f^{\prime} (x) \rightarrow 0$ as 
$x \rightarrow \infty$.

\bigskip

\item{4.3}  Let $f(x,y) = xy - \cos y + x^2 + 1$.  At what points on the set 
$\{(x,y) : f(x,y) = 0 \}$ does the condition $f(x,y) = 0$ fail to define either 
$x$ as a function of $y$ or $y$ as a function of $x$?

\bigskip

\item{4.4}  Prove or give a counterexample: If $f$ is a uniform limit of polynomials 
on $[-1,1]$, then the Maclaurin series of $f$ converges to $f$ in some neighborhood 
of $0$.

\bigskip

\item{4.5}  Can the surface whose equation is
$x y - z \log y + e^{x z} = 1$
 be represented in the form $z = f(x,y)$ in a neighborhood of $(0,1,1)$?  
In the form $y = g(x, z)$ in a neighborhood of $(0,1,1)$?

\bigskip

\item{4.6}  Let $f(x) = z^k \sin (1 \slash x)$ if $x \neq 0$ and $f(0) = 0$.
\medskip
\itemitem{(a)}  If $k = 2$, show that $f$ is differentiable everywhere but $f^{\prime}$ 
fails to be continuous at some point.
\itemitem{b}  If $k = 3$, does $f$ have a second derivative at all points?  If so, is 
$f^{\prime \prime}$ a continuous function?  Give your reasons.

\item{4.7}   Let $f$ be defined on $\RR^3$ by
$f(x,y,z) = x^2 + 4y^2 - 2yz - z^2$.
 Show that $f(2,1,-4) = 0$ and $f_z(2,1,-4) \neq 0$, and that 
there exists therefore a differentiable function $g$ in a neighborhood of $(2,1)$ 
in $\RR^2$, such that $f(x,y,g (x,y)) = 0$.  Find $g_x(2,1)$ and $g_y(2,1)$.  
What is the value of $g(2,1)$?

\bigskip

\item{4.8}  Suppose that a function $f$ is defined on $(0,1]$ and has a finite 
derivative with $| f^{\prime} (x) | < 1$.  Define $a_{n} := f(1 \slash n)$ 
for $n = 1,2,3,\ldots $.  Show that $\lim_{n \to \infty} a_n$ exists.


\bigskip
\item{4.9}  Prove or disprove: The series $\sum_{n = 1}^{\infty} (-1)^n 
{{n^2 + n} \over {n^2}}$ is uniformly convergent on $[-1,1]$.

\item{4.10}  Define a function $f$ on $\RR$ by
$$f(x)=\cases \aligned
e^{-1/x^2}\text{\rm , if $x>0$}\\
0\text{\rm ,  if $x\le 0$     }\endaligned
\\
\endcases $$

\itemitem{(a)} Check whether $f$ is infinitely differentiable at 0, and, if 
so, find $f^{(n)}(0),$ $n=1,2,3,\cdots$.  Show details.
\itemitem{(b)}  Does $f$ have a power series expansion at $0$?
\itemitem{(c)} Let $g(x)=f(x)f(1-x)$.  Show that $g$ is a nontrivial infinitely 
differentiable function on $\RR$ which vanishes outside $(0,1)$.
\bigskip

\item{4.11}  Prove that a function $f : \RR^{n} \to \RR$ is continuous at 
$x = (x_1, \cdots, x_n) \in \RR^{n}$ if the partial derivatives $f_{x_1} , \ldots, f_{x_n} $ exist and are 
bounded in a neighborhood of $x$.

\bigskip

\item{4.12}  Let $f_N(x)~=~\sum_{n = 1}^{N} a_n \sin (nx)$ for $a_n,x \in \RR$.  
If $\sum_{n = 1}^{\infty} na_{n}$ converges absolutely, show that $(f_{N})_{N\in\NN}$ converges 
uniformly to a function $f$ on $\RR$, and that $(f_{N}^{\prime})_{N\in\NN}$ converges 
uniformly to $f^{\prime}$ on $\RR$.

\bigskip

 



\item{4.13}  Let $f$ be a twice continuously differentiable real-valued function on $\RR^n$.  A point $x \in \RR^n$ is a critical point of $f$ 
if all partial derivatives of $f$ vanish at $x$ (i.e., $\bigtriangledown f(x) = 0$), 
a critical point $x$ is nondegenerate if the $n\times n$ matrix
$$\left [{{\partial^2 f} \over {\partial x_i \partial x_j}} (x) \right ]$$
\item{}  is nonsingular.  Let $x$ be a nondegenerate critical point of $
f$.  Prove 
that there is an open neighborhood of $x$ which contains no other critical points.  
(i.e., the nondegenerate critical points are isolated.)

\bigskip

\item{4.14}  Show that the power series in $x$ for the function $f(x) = e^{a x} 
\cos (bx)$ ($a,b\in\RR$) has either no zero coefficients or infinitely many zero coefficients.


\bigskip

\item{4.15}  Let $f$ be continuous function on $[0,1]$.  Define
$$I_{n} = \prod_{j = 1}^{n} \left [ {1 + {1 \over n} f \left ({j \over n} \right )} \right ]$$
\item{}  for every integer $n \geq 1$.  Determine the limit 
$\lim_{n \rightarrow \infty} I_n$.  

\bigskip

\item{4.16}  Show that a function $f(x) = e^{-x} + 2e^{-2x} + \ldots 
+ ne^{-nx} + \ldots$ is continuous on $(0,\infty)$.

\bigskip

\item{4.17}  Let $f(x) = e^{-1 \slash x^2}$ if $x > 0$ and $f(x) = 0$ if 
$x \leq 0$.  Verify, using induction, that for each positive integer $k$, there 
is a polynomial $p_k$ such that $f^{(k)} (x) = f(x) p_k (x) x^{-3k}$ for all 
$x > 0$.  Show that the function $\phi : x \rightarrow f (1 + x) f(1 - x)$ 
has the following properties: 
\medskip
\itemitem{(i)} $\phi$ is smooth (i.e. has derivatives of all orders); 
\itemitem{(ii)} $\phi \geq 0$; 
\itemitem{(iii)} $\phi (x) = 0$ if $| x | \geq 1$; 
\itemitem{(iv)}
$\int_{-\infty}^{\infty} \phi (x) dx > 0$.

\bigskip

\item{4.18}  Let $f : \RR \to \RR$ be a twice differentiable function with 
bounded first and second derivatives.  By considering the Taylor expansion of $f$, 
show that: $ \| f^{\prime}\|_\infty \leq {1 \over h} \| f\|_\infty 
| + h \| f^{\prime \prime} \|_\infty$ for every $h > 0$.  By minimizing over $h$, show that $\| f^{\prime} \|_{\infty} \leq 2 \sqrt{\| f \|_{\infty} \| f^{\prime \prime} \|_{\infty}}$, where $\| g \|_{\infty}$ denotes $\sup_{x \in \RR} 
| g(x) |$.

\bigskip

\item{4.19}  Define the Hermite polynomial of degree $n$ by 
$$H_n (x) = (-1)^n 
e^{{{x^2} \over {2}}} ({{d} \over {dx}} )^n e^{- {{x^2} \over {2}}}\quad ( n \geq 0, x\in\RR).$$
  
Use Taylor's theorem to prove the identity
$$\sum_{n = 0}^{\infty} {{t^n} \over {n!}} H_n (x) = e^{t x - {{1} \over{2}} t^2}.$$
 
\bigskip

\item{4.20}  Let $f$ be a real-valued differentiable function on an interval $(a,b)$. Show that $f$ is Lipschitz continuous if and only if $f$ has bounded derivative.
\vfill
\break

\centerline{\bf {5.  Functions of bounded variation}}
\pni\bigskip\bigskip
\item{5.1}  If $f$ is continuous on an interval $[a,b]$ with a bounded derivative in $(a,b)$, 
show that $f$ is of bounded variation on $[a,b]$.  Is the boundedness of 
$f^{\prime}$ necessary for $f$ to be of bounded variation?  Justify your answer.
\bigskip

\item{5.2}       (a)  Prove that if a real-valued function $f$ is of bounded variation 
on an interval
\itemitem{} $[a,b]$, then $f$ has right and left-hand limits at all $x \in (a,b)$.
\itemitem{(b)}  Prove that a function $f : [a,b] \rightarrow \RR$ 
of bounded variation has at most countably many points of discontinuity.

\item{5.3}  Let $f (x) = x^2 \sin ({{1} \over {x}}), g(x) = x^2 \sin ({{1} \over {x^2}})$ 
for $x \neq 0$ and $f(x) = g(x) = 0$ for $x = 0$.  Show:
\medskip
\itemitem{(a)}  $f$ and $g$ are differentiable everywhere (including $x = 0$),
\itemitem{(b)}  $f$ is bounded variation on the interval $[-1,1]$, but $g$ is not.
\itemitem{(c)}  Let $f(x) = x \sin(1 \slash x)$ for $x \neq 0$ and $f(x) = 0$ for $x = 0$. Is $f$ of bounded variation on $[-1,1]$?
\vfill
\eject


\centerline{\bf {6.  Riemann integral, Lebesgue integral (via completion),}}
\centerline{\bf{ convergence theorems}}
\pni\bigskip\bigskip
\item{6.1}  Let $f$ be a positive function on $(0,1]$ such that $f$ is Riemann 
integrable on $[t,1]$ for all $t\in (0,1)$, but $\lim_{x \to 0^{+}} f(x) = \infty$.  Assume that the improper (Riemann) integral $(R) 
\int_{0}^{1} f(x) dx$ exists.  Show that $f$ is a measurable function, 
Lebesgue integrable, and
$$\int_{[0,1]} f(x) dx~=~(R) \int_{0}^{1} f(x) dx.$$

\bigskip

\item{6.2}  For each of the following problems, check whether the limit exists.  
If so, find its value.
$${\rm (a)} \lim_{n \rightarrow \infty} \int_{1}^{n} ({1 - {x \over n}})^n 
dx,~~~~{\rm (b)} \lim_{n \rightarrow \infty} \int_{1}^{2 n} ({1 - {x \over n}} 
)^n dx.$$

\bigskip

\item{6.3}   (a)   Characterize those bounded functions 
on $[0,1]$ which are Riemann integrable.
\itemitem{(b)}  Let $({r_n})$ be an enumeration of the rationals in $[0,1]$.  
Define $f$ on $[0,1]$ by
$$f(x)~=\left\{\aligned
\frac 1n\text{\rm ,   if $x=r_n$       }\\
0\text{\rm , if $x$ is irrational}\endaligned
\right.$$
Is $f$ Riemann integrable on [0,1]?  Explain!
\itemitem{(c)}  Show that if
$$f(x)~=~\left\{\aligned
1\text{\rm , if x is rational  }\\
-1\text{\rm , if x is irrational}\endaligned
\right.$$
then $f$ is not Riemann integrable on the interval $[0,1]$.  Is $
f$ 
Lebesgue integrable?  Explain!

\bigskip

\item{6.4}  Show there are no bounded sequences $(a_n)_{n\in\NN}$ and $(b_n)_{n\in\NN}$ for 
which $f_n (x) = a_n \sin (2 \pi n x) + b_n \cos (2 \pi n x )$ converges to 1 
almost everywhere on $[0,1]$.

\bigskip

\item{6.5}  Let $f(x)$ be a real-valued measurable function on $[0,1]$.  Show that
$$\lim_{n \rightarrow \infty} \int_{0}^{1} ( \cos (\pi f(x)))^{2 n} dx = m 
\{x: f(x) \,\,{\rm is~a n~integer} \},$$
 where $m$ denotes Lebesgue measure.

\bigskip

\item{6.6}    (a)  Show that $f(x) = x^{-r}$ is a Lebesgue integrable function on 
$[0,1]$ if $0 \leq r < 1$.
\itemitem{(b)}  If $0 \leq r < 1$ let $\displaystyle{a_n = \int_{0}^{1} {{dx} \over {n^{-1} 
+ x^{r}}}}$ (Lebesgue integral).  Compute $\lim_{n \rightarrow \infty} a_n$.  Be 
sure to quote the appropriate integration theorems to justify your calculations.

\bigskip

\item{6.7}  Let $f_n : I\hskip-.13cmR \rightarrow I\hskip-.13cmR$ be defined 
by
$$f_n(x)=\left\{\aligned
\frac 1n\text{\rm , if $\vert$$x\vert\le n$}\\
0\text{\rm , if $\vert x\vert >n$}\endaligned
\right.$$
\itemitem{(a)}  Show that $f_n$ converges to $0$ uniformly on 
\RR,  and that $\lim_{n\rightarrow\infty}\int_{\RR}f_n(x)\,dx=2$ 
while $\int_{\RR}\left(\lim_{n\rightarrow\infty}f_n\right)(x)\,dx
\neq 2$.
\itemitem{(b)}  Explain why the example in part (a) does not contradict the 
Lebesgue dominated convergence theorem.

\bigskip

\item{6.8}   (a) Show that $f(x) = 1 \slash \sqrt{x}$ is Lebesgue integrable on 
$(0,1)$.
\itemitem{(b)}  Find $\inf \{\int_{0}^{1} \psi(x) \, dx | \psi~{\rm is~a~simple~function,~
a n d}\,\, \psi (x) \geq 1 \slash \sqrt{x}~{\rm o n}~(0,1) \}$.  (Simple functions are 
finite linear combinations of characteristic functions of measurable sets with 
extended real-valued coefficients.)

\bigskip

\item{6.9}  Give an example of a sequence $(f_n)_{n\in\NN}$ of bounded, measurable 
functions on $[0,1)$ such that
$$\lim_{n \rightarrow \infty} \int_{0}^{1} | f_n (x) | \, dx = 0$$
 but such that $f_n$ converges pointwise  nowhere.

\bigskip

\item{6.10}  Consider a Lebesgue-measurable function $f$ or {{I\hskip-.13cmR}} with 
$\int \limits_{{I\hskip-.13cmR}} f(t)^2 \, dt < \infty$.  show that the function $g(x) =
 \int \limits_{{I\hskip-.13cmR}} f(t - x) f(t) \, dt$ is continuous.


\bigskip

\item{6.11}  Prove that $\lim_{n \rightarrow \infty} \int_{- \infty}^{\infty} 
(\sin nt) f(t) dt~=~0$ for every Lebesgue integrable function $f$ or {\bf R}.  
(Hint:  Do the problem first for step functions.)

\bigskip

\item{6.12}  Let $\displaystyle{f_n (x) = {{n} \over {x (ln x)^n}}}$ for $x \geq e$ and 
$n \in {\NN}$.
\itemitem{(a)}  For which  $n \in  {\NN}$ does the Lebesgue integral $\int 
\limits_{e}^{\infty} f_n (x) \, dx$ exist?
\itemitem{(b)}  Determine $\lim_{n \to \infty} f_n (x)$ for $x > e$.
\itemitem{(c)}  Does the sequence $( f_n )_{n\in\NN}$ satisfy the assumptions of 
Lebesgue's dominated convergence theorem?

\bigskip

\item{6.13}  Define $f(x) = \int_{\RR} \cos (xy) g(y) \, dy$ for $x \in  
{I\hskip-.13cmR}$ where $g$ is an integrable function on ${I\hskip-.13cmR}$.  Show that $f$ is 
continuous.

\bigskip

\item{6.14}  Define $g : I\hskip-.13cmR \rightarrow I\hskip-.13cmR$ by 
$g(x) = 0$ if $x$ is irrational and $g(x) = {1 \over q}$ if $x = p \slash q$ 
in lowest terms.  Is $g$ a Riemann integrable function?  Give a proof of your 
assertion.

\bigskip

\item{6.15}  Give an example of a Lebesgue integrable function $f$ on 
$[0,1]$ such that $\int_{0}^{1} f(x) \, dx = 1$, but $f$ is not Riemann integrable.

\bigskip

\item{6.16}   Let $f \in L^{\infty} [0,1]$ and $\int_{0}^{1} x^n 
f(x)\, dx = 0$ for $n \in \NN$.  Show that $f = 0$ a.e.

\bigskip


\item{6.17}  Let $f$ be a non-negative Lebesgue measurable function on 
$(0, \infty)$ such that $f^2$ is integrable.  Let $F(x) = \int_{0}^{x} f(t) 
\, dt$ where $x > 0$.  Show that
$\lim_{x \rightarrow 0^+} {{F(x)} \over {\sqrt{x}}} = 0.$

\bigskip

\item{6.18}  Let $f$ be a differentiable function on $[-1,1]$.  Prove that 
$\lim_{\epsilon \to 0} \int_{\epsilon < | x | \leq 1} {1 \over x} 
f(x)\, dx$ exists.

\bigskip

\item{6.19}  Let $\displaystyle{f_n (x) : = {{x^n} \over {n!}} e^{-x}}$ for $n \in
 {\NN}_0 $.
\itemitem{(a)}  Show that $\lim_{x \rightarrow \infty} f_n (x) = 0$ for 
all $x > 0$.
\itemitem{(b)}  Show that $f_n \in L^1 (0, \infty)$ with $\| 
f_n \|_1 = 1$ for 
all $n \in {\NN}_0$.
\itemitem{(c)}  Show that $\lim_{k\rightarrow\infty}\int_0^k\frac {
x^n}{n!}(1-\frac xk)^k\,dx=1$ for 
all $n\in\NN_0$.

\bigskip

\item{6.20}  Prove that
$\lim_{b \rightarrow \infty} \int_{0}^{b} {{\sin x} \over {x}}\, dx$
 exists but that the function ${{\sin x} \over x}$ is not integrable 
over $(0, \infty)$.

\bigskip

\item{6.21}  Compute the following limit and justify your calculations:
$$\lim_{n \rightarrow \infty} \int_{0}^{\infty} \left (1 + {{x} \over {n}} 
\right )^{-n} \sin \left ( {x \over n} \right )\, dx$$

\bigskip

\item{6.22}  Let $f$ be a continuous nonnegative function on $[a,b]$ where 
$a < b$.  Let $M = \max \{f(x) : a \leq x \leq b \}$.  Show that
$$\lim_{n \rightarrow \infty} \left (\int_{a}^{b}  f(x)^n \, dx \right )^{{1} \over {n}}
 = M.$$

\bigskip

\item{6.23}  Find and justify the limits: (a) $\displaystyle{\lim_{n \rightarrow \infty} 
\int_{0}^{n} {{\sin x} \over {1 + nx^2}} \, dx}$ and (b) $\displaystyle{\lim_{n \rightarrow \infty} 
    \int_{0}^{e^n} {x \over {1 + nx^2}} \, dx}$.

\bigskip


\item{6.24}  Let $f_n(x) = \sum_{i = 0}^{n - 1} {1 \over n} f (x + {i \over n})$, 
where $f$ is a continuous function on $I\hskip-.13cmR$.  Show that the sequence 
of functions $(f_n)_{n\in\NN}$ converges uniformly on every finite 
segment $[a,b]$ to the function $F(x) = \int_{x}^{x + 1} f(s)\, ds$.

\bigskip

\item{6.25}  Let $f \in L^1 (I\hskip-.13cmR)$  with respect to the 
Lebesgue measure such that $\int_{\RR} | x | | f(x) | \, dx < \infty$.  
Show that the function
$$g(y) = \int_{\RR} e^{ixy} f(x) dx$$
 is differentiable at every $y \in I\hskip-.13cmR$.

\bigskip

\item{6.26}  Prove that if $f$ is a real-valued Lebesgue integrable function on $
{I\hskip-.13cmR}$, then
$$\lim_{x \rightarrow 0} \int | f(x + t) - f(t) | \, dt = 0.$$

\bigskip

\item{6.27}  Let $f \in L^1 ({I\hskip-.13cmR}).$
\itemitem{(a)}  Prove:  $\lim_{n \rightarrow \infty} \int_{0}^{1 \slash n} f(x) 
dx = 0.$
\itemitem{(b)}  Prove or disprove: $\lim_{n \rightarrow \infty} \int_{n}^{\infty} f(x) \, dx 
= 0.$

\bigskip

\item{6.28}  Give an example of a sequence of uniformly bounded measurable 
functions $f_n$ on $[0,1]$ such that $m \{ x | f_n (x) \neq 0 \} 
\rightarrow 0$ as $n \rightarrow \infty$, but the sequence $f_n (x)$ does 
not converge for any $x \in [0,1].$

\bigskip

\item{6.29}  Let
$$f_n(x)=\left\{\aligned
n^2x\text{\rm ,    for $0\le x<\frac 1n$  }\\
2n-n^2x\text{\rm , for $\frac 1n\le x\le\frac 2n$}\\
0\text{\rm ,       for $\frac 2n<x\le 1$ }\endaligned
\right.$$
Sketch the graphs of $f_1$ and $f_2$.  Prove that if $g$ is a continuous 
real-valued function on $[0,1]$, then
$$\lim_{n \rightarrow \infty} \int_{0}^{1} f_n (x) g (x) dx = g (0).$$
(Hint:  First show that $\int_0^1f_n(x)\,dx=1$.)

\bigskip

\item{6.30}  Assume that the real valued measurable function $f(t,x)$ and its 
partial derivative ${{\partial} \over {\partial t}} f (t,x)$ are bounded on $[0,1]^2$.  
Show that for $t \in (0,1)$
$${{d} \over {d t}} \left [{\int_{0}^{1} f(t,x) \, dx} \right ] = \int_{0}^{1} {{\partial} 
\over {\partial t}} f (t,x) \, dx .$$
 Hint:  Consider the difference quotient for the derivative on the left.

\bigskip


\item{6.31}  Prove that if $f_n$ is Lebesgue integrable on $[0,1]$ for each $n\in\NN$, 
and $\sum_{n = 1}^{\infty} \int_{0}^{1}  | f_n (x) | \, dx < \infty$, then 
$\sum_{n = 1}^{\infty} f_n (x)$ is convergent { a.e.}, and
$$\int_{0}^{1} \sum_{n = 1}^{\infty} f_n (x) \, dx = \sum_{n = 1}^{\infty} \int_{0}^{1} f_n (x) dx.$$

\bigskip

\item{6.32}  Let $f \in L^1 (I\hskip-.13cmR)$.  Prove that $\lim_{n 
\rightarrow \infty} \int_{0}^{n} e^{-n x} f(x) \, dx = 0.$

\bigskip


\item{6.33}  Assume that $A \geq 0$, $B > 0$, and $f$ continuous and 
nonnegative on $[a,b]$.  Assume that $f(t) \leq A + B \int_{a}^{t} 
f(x)\,  ds$ for $a \leq t \leq b$.  Prove that $f(t) \leq A e^{B (t - a)}$ for 
$a \leq t \leq b$.
\vfill
\eject
\centerline{\bf {7. Absolutely continuous functions and the fundamental}}
\centerline{\bf{ theorem of calculus}}
\pni\bigskip\bigskip

\item{7.1}  Show that  the product of two absolutely continuous functions on 
a closed finite interval $[a,b]$ is absolutely continuous.

\bigskip

\item{7.2}    (a)  Show that a Lipschitz function is absolutely continuous.
\itemitem{(b)}  Show that an absolutely continuous function $f$ on an interval is Lipschitz if 
and only if $f^{\prime}$ is bounded.

\bigskip

\item{7.3}  Let $f$ be absolutely continuous in the interval $[\epsilon, 1]$ 
for each $\epsilon > 0$.  Does the continuity of $f$ at $0$ imply that $f$ is 
absolutely continuous on $[0,1]$?  What if $f$ is also of bounded variation 
on $[0,1]$?

\bigskip

\item{7.4}  A function $f : [0,1] \to L^1  [0,1]$ is called Lipschitz 
continuous if there exists $M > 0$ such that $\| f(t) - f (s)\|_1 
\leq M |t - s|$ for all $t, s \in [0,1]$.  It is called differentiable 
at a point $s \in (0,1)$ if the differential quotients ${f (t) - f (s)\over (t - s)}$ converge in $L^1 [0,1]$ as $t \to s$.  Let $f : [0,1] \to  L^1 [0,1]$ be given by $f (t) = {\chi}_{[0,t]}$, where ${\chi}_{[0,t]}$ 
denotes the characteristic function of the interval $[0,t]$.  Show that $f$ is Lipschitz 
continuous and nowhere differentiable.
\vfill
\eject
\centerline{\bf {8. Basic properties of $L^p$-spaces, Riesz representation for 
$L^p$-spaces}}
\pni\bigskip\bigskip

\item{8.1}  Show that $(L_p [0,1],\| \cdot \|_p)$ is separable for $1 \leq p < \infty$, but not separable for $p = \infty$.

\bigskip

\item{8.2}  Show that $L^p (0,1) \subset L^q (0,1)$ for any $p > q \geq 1$.  
Here the integrability is with respect to the Lebesgue measure.  Is the inclusion 
map for $L^p (0,1)$ into $L^q (0,1)$ continuous?

\bigskip

\item{8.3}  Prove or disprove the equality $\displaystyle{L^{\infty} [0,1] = \cap_{1 \leq 
p < \infty} L^p [0,1]}$.

\bigskip

\item{8.4}  Let $f \in L_p (I\hskip-.13cmR), 1 \leq p < \infty$.  Show 
that
$\displaystyle{\int_{| x | > n} | f (x) |^p \, dx \rightarrow 0}$  for $n \rightarrow \infty.$

\bigskip

\item{8.5}  Let $g_n = n \chi_{[0,n^{-3}]}$.  Show that
$\int_0^1 f(x) g_n(x)\, dx \rightarrow 0$ as $n\to\infty$ for all $f \in L^2 [0,1]$, 
  but not all $f \in  L^1 [0,1]$.

\bigskip




\item{8.6}  Construct an isometry of the Hilbert space $\ell_2$ onto the Hilbert 
space $L_2~[0,1 ]$ and justify that your map is an isometry.

\vfill
\bye




\bye


\centerline{\bf {6.  Riemann integral, Lebesgue integral (via completion), convergence theorems}}
\pni
\item{6.1}  Let $f$ be a positive function on $(0,1]$ such that $f$ is Riemann 
integrable on $[t,1]$ for all $t~\in~(0,1)$, but $\lim_{x \rightarrow 0^{+}} f(x) = \infty$.  Assume that the improper (Riemann) integral $(R) 
\int_{0}^{1} f(x) dx$ exists.  Show that $f$ is a measurable function and is 
Lebesgue integrable and
$$\int_{[0,1]} f(x) dx~=~(R) \int_{0}^{1} f(x) dx.$$

\item{6.2}  For each of the following problems, check whether the limit exists.  
If so, find its value.
$${\rm (a)} \lim_{n \rightarrow \infty} \int_{1}^{n} ({1 - {x \over n}})^n 
dx,~~~~{\rm (b)} \lim_{n \rightarrow \infty} \int_{1}^{2 n} ({1 - {x \over n}} 
)^n dx.$$

\item{6.3}    \hfill
\itemitem{a}  Give a necessary and sufficient condition for a bounded function 
on $[0,1]$ to be Riemann integrable.
\itemitem{b}  Let $({r_n})$ be an enumeration of the rationals in $[0,1]$.  
Define $f$ on $[0,1]$ 
by
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases{
1 \slash n & if $x = r_n$ \cr
0 & if $x$ is irrational
}
$$\itemitem{}Is$$

f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$ Riemann integrable on $[0,1]$?  Explain!
\itemitem{c}  Show that if
$$f(x)~=~
\left \{
\eqalign{1& x {\rm rational} \cr
                 -1& x {\rm irrational} \cr}
\right \}
$$
\itemitem{}  then $f$ is not Riemann integrable on any interval $[0,1]$.  Is $f$ 
Lebesgue integrable?  Explain!

\item{6.4}  Show there are no bounded sequences $\{a_n \}$ and $\{b_n \}$ for 
which $f_n (x) = a_n \sin (2 \pi n x) + B_n \cos (2 \pi n x )$ converges to 1 
almost everywhere on $[0,1]$.

\item{6.5}  Let $f(x)$ be a real-valued measurable function on $[0,1]$.  Show 
$\lim_{n \rightarrow \infty} \int_{0}^{1} ( \cos (\pi f(x)))^{2 n} dx = m 
\{x: f(x) {\rm is~a n~interger} \}$ where $m$ denotes Lebesgue measure.

\item{6.6}    \hfill
\itemitem{a}  Show that $f(x) = x^{-r}$ is a Lebesgue integrable function on 
$[0,1]$ if $0 < r < 1$.
\itemitem{b}  If $0 < r < 1$ let $a_n = \int_{0}^{1} {{dx} \over {n^{-1} 
+ x^{r}}}$ (Lebesgue integral).  Compute $\lim_{n \rightarrow \infty} a_n$.  Be 
sure to quote the appropriate integration theorems to justify your calculations.

\item{6.7}  Let $f_n : \RR \rightarrow \RR$ be defined 
by
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x) = \cases{
$1 \over n$ & if $| x | \leq n$ \cr
0 & if $| x | > n$. \cr
}
$$\itemitem{a}Showthat$$

f
f
f
f
f
f
f_n$ converges to $0$ uniformly on $\RR$ 
and that $\lim_{n \rightarrow \infty} \int_{\RR} f_n (x) dx = 2$ 
while $\int_{\RR} \left (\lim_{n \rightarrow \infty} f_n \right ) 
(x) dx \neq 2$.
\itemitem{b}  Explain why the example in part (a) does not contradict the 
Lebesgue dominated convergence theorem.

\item{6.8}     \hfill
\itemitem{a}  Show that $f(x) = 1 \slash \sqrt{x}$ is Lebesgue integrable on 
$(0,1)$.
\itemitem{b}  Find $I n f \{\int_{0}^{1} \psi | \psi~{\rm is~a~simple~function,~
a n d} \psi (x) \geq 1 \slash \sqrt{x}~{\rm o n}~(0,1) \}$.  (Simple functions are 
finite linear combinations of characteristic functions of measurable sets with 
extended real-valued coefficients.)

\item{6.9}  Give an example of a sequence $\{f_n \}$ of bounded, measurable 
functions on $[0,1)$ such that
$$\lim_{n \rightarrow \infty} \int_{0}^{1} | f_n | = 0$$
\item{}  but such that $f_n$ converges pointwise {\it nowhere}.

\item{6.10}  Consider a Lebesgue-measurable function $f$ or {\bf R} with 
$\int \limits_{\bf R} f(t)^2 dt < \infty$.  show that the function $g(x) =
 \int \limits_{\bf R} f(t - x) f(t) dt$ is continuous.
\item{6.11}  Prove that $\lim_{n \rightarrow \infty} \int_{- \infty}^{\infty} 
(\sin nt) f(t) dt~=~0$ for every Lebesgue integrable function $f$ or {\bf R}.  
(Hint:  Do the problem first for step functions.)

\item{6.12}  Let $f_n (x) = {{n} \over {x (ln x)^n}}$ for $x \geq e$ and 
$n \in N$.
\itemitem{a}  For which  $n \in N$ does the Lebesgue integral $\int 
\limits_{e}^{\infty} f_n (x) dx$ exist?
\itemitem{b}  Determine $\lim_{n \rightarrow \infty} f_n (x)$ for $x > e$.
\itemitem{c}  Does the sequence $\{ f_n \}$ satisfy the assumptions of 
Lebesgue's dominated convergence theorem?

\item{6.13}  Define $f(x) = \int_{\bf R} \cos (xy) g(y) dy$ for $x \in$ 
{\bf R} where $g$ is an integrable function on {\bf R}.  Show that $f$ is 
continuous.

\item{6.14}  Define $g : \RR \rightarrow \RR$ by 
$g(x) = 0$ if $x$ is irrational and $g(x) = {1 \over q}$ if $x = p \slash q$ 
in lowest terms.  Is $g$ a Riemann integrable function?  Give a proof of your 
assertion.

\item{6.15}  Give an example of a Lebesgue integrable function $f$ on 
$[0,1]$ such that $\int_{0}^{1} f(x) dx = 1$, but $f$ is not Riemann integrable.

\item{6.16}      \hfill
\itemitem{a}  Let $f$ be in $L_{\infty} [0,1]$ such that $\int_{0}^{1} x^n 
f(x) dx = 0$ for $n = 1, 2, 3, \ldots$.  Show that $f \equiv 0$.
\itemitem{b}  If, in part (a), $f$ is not continuous but just a bounded 
Lebesgue measurable function, show that $f = 0$ {\it a.e.}

\item{6.17}  Let $f$ be a non-negative Lebesgue measurable function on 
$(0, \infty)$ such that $f^2$ is integrable.  Let $F(x) = \int_{0}^{x} f(t) 
dt$ where $x > 0$.  Show that
$$\lim_{x \rightarrow 0+} {{F(x)} \over {\sqrt{x}}} = 0.$$

\item{6.18}  Let $f$ be a differentiable function on $[-1,1]$.  Prove that 
the following limit exists
$$\lim_{\in \rightarrow 0} \int_{\in < | x | \leq 1} {1 \over x} 
f(x) dx.$$

\item{6.19}  Let $f_n (x) : = {{x^n} \over {n!}} e^{-x}$ for $n \in 
N_0 : = \{0, 1, 2, 3\ldots \}$.
\itemitem{a}  Show that $\lim_{x \rightarrow \infty} f_n (x) = 0$ for 
all $x > 0$.
\itemitem{b}  Show that $f_n \in L^1 (0, \infty)$ with $\| 
f_n \|_1 = 1$ for 
all $n \in N_0$.
\itemitem{c}  Show that $\lim_k \rightarrow \infty \int_{0}^{k} {{x^n} 
\over {n!}} (1 - {{x} \over {k}} )^k dx = 1$ for all $n \in N_0$.

\item{6.20}  Prove that
$$\lim_{b \rightarrow \infty} \int_{0}^{b} {{\sin x} \over {x}} dx$$
\item{}  exists but that the function ${\sin x} \over x$ is not integrated 
in $(0, \infty)$.

\item{6.21}  Compute the following limit and justify your calculations:
$$\lim_{n \rightarrow \infty} \int_{0}^{infty} \left (1 + {{x} \over {n}} 
\right )^{-n} \sin \left ( {x \over n} \right ) dx$$

\item{6.22}  Let $f$ be a continuous nonnegative function on $[a,b]$ where 
$a < b$.  Let $M = max \{f(x) : a \leq x \leq b \}$.  Show
$$\lim_{n \rightarrow \infty} \left (\int_{a}^{b}  f(x)^n dx \right )^{{1} \over {n}}
 = M.$$

\item{6.23}  Find (and justify) the limits (a) $\lim_{n \rightarrow \infty} 
\int_{0}^{n} {{\sin x} \over {1 + nx^2}} dx$ and (b) $\lim_{n \rightarrow \infty} 
\int_{0}^{e^n} {x \over {1 + nx^2}} dx$.

\item{6.24}  Let $f_n(x) = \sum_{i = 0}^{n - 1} {1 \over n} f (x + {i \over n})$, 
where $f$ is a continuous function on $\RR$.  Show that the sequence 
of functions $\left\langle  f_n \right\rangle$ converges uniformly on every finite 
segment $[a,b]$ to the function $F(x) = \int_{x}^{x + 1} f(s) ds$.

\item{6.25}  Let $f \varepsilon L^1 (\RR)$  with respect to the 
Lebesgue measure such that $\int_R | x | | f(x) | dx < \infty$.  
Show that the following function
$$g(y) = \int_R e^{fxy} f(x) dx$$
\item{}differentiable at every $y \varepsilon \RR$.

\item{6.26}  Prove that if $f$ is a real-valued Lebesgue integrable function on 
{\bf R}, then
$$\lim_{x \rightarrow 0} \int | f(x + t) - f(t) | dt = 0.$$

\item{6.27}  Let $f \in L^1 ({\bf R}).$
\itemitem{a}  Prove:  $\lim_{n \rightarrow \infty} \int_{0}^{1 \slash n} f(x) 
dx = 0.$
\itemitem{b}  Must $\lim_{n \rightarrow \infty} \int_{n}^{\infty} f(x) dx 
= 0.$

\item{6.28}  Give an example of a sequence of uniformly bounded measurable 
functions $f_n (x)$ on $[0.1]$ such that $m \{ x | f_n (x) \neq 0 \} 
\rightarrow 0$ as $n \rightarrow \infty$, but the sequence $\{f_n (x) \}$ does 
not converge for any $x \in [0,1].$

\item{6.29}  Let
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n (x) = \cases{
$4n^2x$, & for $0 \leq x \leq 1 \slash (2n)$; \cr
$2n - n^2$ $(x - 1 \slash n)$, & for $1 \slash n \leq x \leq 2 \slash n$; \cr
$0$, & for $2 \slash n \leq x \leq 1$. \cr
}
$$
\item{} Sketch the graphs $







f_1$ and $f_2$.  Prove that if $g$ is a continuous 
real-valued function on $[0,1]$, then.
$$\lim_{n \rightarrow \infty} \int_{0}^{1} f_n (x) g (x) dx = g (0).$$
\item{}  {\bf Hint:}  First show that $\int_{0}^{1} f_n dx = 1.$

\item{6.30}  Assume that the real valued measurable function $f(t,x)$ and its 
partial derivative ${{\partial} \over {\partial t}} f (t,x)$ are bounded on $[0,1]^2$.  
Show that for $t \in (0,1)$
$${{d} \over {d t}} \left [{\int_{0}^{1} f(t,x) dx} \right ] = \int_{0}^{1} {{\partial} 
\over {\partial t}} f (t,x) dx$$
\item{}({\bf HINT:}  Consider the difference quotient for the derivative on the left.)

\item{6.31}  Prove that if $f_n$ is Lebesgue integrable on $[0,1]$ for each $n$, 
and $\sum_{n = 1}^{\infty} \int_{0}^{1}  | f_n (x) | dx < \infty$, then 
$\sum_{n = 1}^{\infty} f_n (x)$ is convergent {\it a.e.}, and
$$\int_{0}^{1} \sum_{n = 1}^{\infty} f_n (x) dx = \sum_{n = 1}^{\infty} \int_{0}^{1} f_n (x) dx.$$

\item{6.32}  Let $f \in L^1 (\RR)$.  Prove that $\lim_{n 
\rightarrow \infty} \int_{0}^{n} e^{-n x} f(x) dx = 0.$

\item{6.33}  Assume that $A \geq 0$, $B > 0$, and $f$ continuous and 
nonnegative on $[a,b]$.  Assume that $f(t) \leq A + B \int_{a}^{t} 
f(x) ds$ for $a \leq t \leq b$.  Prove that $f(t) \leq A e^{B (t - a)}$ for 
$a \leq t \leq b$.
\vfill
\eject
\centerline{\bf {7. Absolutely continuous functions and the fundamental theorem of calculus}}
\pni
\item{7.1}  Show that  the product of two absolutely continuous functions on 
a closed finite interval $[a,b]$ is absolutely continuous.

\item{7.2}     \hfill
\itemitem{a}  Show that a Lipschitz function is absolutely continuous.
\itemitem{b}  Show that an absolutely continuous function $f$ is Lipschitz if 
and only if $f^{\prime}$ is bounded.

\item{7.3}  Let $f$ be absolutely continuous in the interval $[\in, 1]$ 
for each $\in > 9$.  Does the continuity of $f$ at $0$ imply that $f$ is 
absolutely continuous on $[0,1]$?  what if $f$ is also of bounded variation 
on $[0,1]$?

\item{7.4}  A function $f : [0,1] \rightarrow L^1  [0,1]$ is called Lipschitz 
continuous if there exists $M > 0$ such that $\| f(t) - f (s) \|_1 
\leq M | t - s |$ for all $t, s \in [0,1]$.  It called differentiable 
at a point $s \in (0,1)$ if the differential quotients ${{f (t) - f (s)} \over 
{t~-~s}}$ converge in $L^1 [0,1]$ as $t \rightarrow s$.  Let $f : [0,1] 
\rightarrow L^1 [0,1]$ be given by $f (t) = \chi_{[0,t]}$, where $\chi_{[0,t]}$ 
denotes the characteristic function of the interval $[0,t]$.  Show that $f$ is Lipschitz 
continuous and nowhere differentiable.
\vfill
\eject
\centerline{\bf {8. Basic properties of $L^p$-spaces, Riesz representation for 
$L^p$-spaces}}
\pni
\item{8.1}  Show that $(L_p [0,1],\| \cdot \|_p)$ is separable for $1 \leq p < \infty$, but not separable for $p = \infty$.

\item{8.2}  Show that $L^p (0,1) \subset L^q (0,1)$ for any $p > 1 \geq 1$.  
Here the integrability is with respect to the Lebesgue measure.  Is the inclusion 
map for $L^p (0,1)$ into $L^q (0,1)$ continuous?

\item{8.3}  Prove or disprove the equality $L^{\infty} [0,1] = \cap_{1 \leq 
p < \infty} L^p [0,1]$.

\item{8.4}  Let $f \in L_p (\RR), 1 \leq p < \infty$.  Show 
that
$$\int_{| x | > n} | f (x) |^p dx \rightarrow 0~{\rm f o r}~n \rightarrow \infty.$$

\item{8.5}  Let $g_n = n \chi_{[0,n^{-3}]}$.  Show
$$\int f g + n \rightarrow 0~~~~{\rm f o r~a l l} f \in L^2 [0,1],$$
\item{}  but not all $f \in L^1 [0,1]$.


\vfill
\bye

\itemitem{c}  Show that the set $\{f \in L_ {\alpha} | \| f 
\|_{\alapha} \geq 1, f(a) = o \}$ is compact in $C([a, b])$ (with the 
supremum norm).   {\bf 3}



\item{1.2}  Let $(E, \|\cdot\|)$ be a Banach space and $F$ a finite dimensional subspace.  
Show that $(F, \|\cdot\|)$ is a Banach space.

\itemitem{(c)} Let $C[0,1]$ denote the Banach space of all continuous real functions on $
[0,1]$ equipped with the sup-norm and $C_0[0,1]$ the closed subspace of $
C[0,1]$ consisting of those $f$ with $f(0)=0$. Let $R[0,1]$ denote the completion of $
X$. Consider the antiderivative operator $I:C[0,1]\to C_0[0,1]$ defined by $
If=F$, where $F(t)=\int_0^tf(s)\,ds$. Show that $I$ extends to an isometric isomorphism between $
R[0,1]$ and $C_0[0,1]$.
\end
