A sequence $\{s_n\}$ {\it converges to a limit $s$\/} if for every  $\epsilon>0$ there is an integer $N$ such that \begin{equation}\label{eq:4.1.2} |s_n-s|<\epsilon\mbox{\quad if\quad} n\ge N. \end{equation} In this case we say that $\{s_n\}$ is {\it convergent\/} and write $$ \lim_{n\to\infty}s_n=s. $$ A sequence that does not converge {\it diverges\/}, or is {\it divergent\/} \bbox


A sequence $\{s_n\}$ is  {\it bounded above\/}  if there is a real number $b$ such that $$ s_n\le b\mbox{\quad for all $n$}, $$ {\it bounded below\/} if there is a real number $a$ such that $$ s_n\ge a\mbox{\quad for all $n$}, $$ or {\it bounded\/} if there is a real number $r$ such that $$ |s_n|\le r\mbox{\quad for all $n$}. $$


A sequence $\{s_n\}$ is {\it nondecreasing\/} if $s_n\ge s_{n-1}$ for all $n$, or {\it nonincreasing\/} if $s_n\le s_{n-1}$ for all $n.$ A {\it monotonic sequence\/} is a sequence that is either nonincreasing or nondecreasing. If $s_n>s_{n-1}$ for all $n$, then $\{s_n\}$ is  {\it increasing\/}, while if $s_n<s_{n-1}$ for all $n$, $\{s_n\}$ is {\it decreasing\/}.


The numbers $\overline{s}$ and $\underline{s}$ defined in Theorem~\ref{thmtype:4.1.9} are called the {\it limit superior\/} and {\it limit inferior\/}, respectively, of $\{s_n\}$, and denoted by $$ \overline{s}=\limsup_{n\to\infty} s_n\mbox{\quad and\quad} \underline{s}=\liminf_{n\to\infty} s_n. $$ We also define \begin{eqnarray*} \limsup_{n\to\infty}s_n\ar=\phantom{-}\infty\quad\mbox{if $\{s_n\}$ is not bounded above},\\ \limsup_{n\to\infty}s_n\ar=-\infty\quad\mbox{if }\lim_{n\to \infty}s_n=-\infty,\\ \liminf_{n\to\infty} s_n\ar=-\infty\quad\mbox{if $\{s_n\}$ is not bounded below},\\ \arraytext{and}[2\jot] \liminf_{n\to\infty} s_n\ar=\phantom{-}\infty\quad\mbox{if } \lim_{n\to\infty} s_n=\infty. \end{eqnarray*}


A sequence $\{t_k\}$ is  a {\it subsequence\/} of a sequence $\{s_n\}$ if $$ t_k=s_{n_k},\quad k\ge0, $$ where $\{n_k\}$ is an increasing infinite sequence of integers in the domain of $\{s_n\}$.  We denote the subsequence $\{t_k\}$ by $\{s_{n_k}\}$. \bbox


If $\{a_n\}_k^\infty$ is an infinite sequence of real numbers, the symbol $$ \sum_{n=k}^\infty a_n $$ \newpage \noindent is an {\it infinite series\/}, and $a_n$ is the {\it $n$th term\/} of the series. We say that $\sum_{n=k}^\infty a_n$ {\it converges to the \enlargethispage{\baselineskip} sum $A$\/},  and write $$ \sum_{n=k}^\infty a_n=A, $$ if the sequence $\{A_n\}_k^\infty$ defined by $$ A_n=a_k+a_{k+1}+\cdots+a_n,\quad n\ge k, $$ converges to $A$. The finite sum $A_n$ is  the {\it $n$th partial sum of\/} $\sum_{n=k}^\infty a_n$. If $\{A_n\}_k^\infty$ diverges, we say that $\sum_{n=k}^\infty a_n$ {\it diverges\/};  in particular, if $\lim_{n\to\infty}A_n =\infty$ or $-\infty$, we say that $\sum_{n=k}^\infty a_n$ {\it diverges to $\infty$ or $-\infty$\/}, and write $$ \sum_{n=k}^\infty a_n=\infty\mbox{\quad or\quad}\sum_{n=k}^\infty a_n= -\infty. $$ A divergent infinite series that does not diverge to $\pm\infty$ is said to {\it oscillate\/}, or {\it be oscillatory\/}. \bbox


{\it converges absolutely\/}, or is {\it  absolutely convergent\/}$,$ if $\sum|a_n|<\infty.$


The {\it Cauchy product of $\sum_{n=0\/}^\infty a_n$ and\/} $\sum_{n=0}^\infty b_n$ is $\sum_{n=0}^\infty c_n$, where \begin{equation}\label{eq:4.3.38} c_n=a_0b_n+a_1b_{n-1}+\cdots+a_{n-1}b_1+a_nb_0. \end{equation} Thus, $c_n$ is the sum of all products $a_ib_j$, where $i\ge0$, $j\ge0$, and $i+j=n$; thus, \begin{equation} \label{eq:4.3.39} c_n=\sum_{r=0}^n a_rb_{n-r}=\sum_{r=0}^n b_ra_{n-r}. \end{equation}


Suppose that  $\{F_n\}$ is a sequence of functions on $D$ and the sequence of values $\{F_n(x)\}$ converges for each $x$ in some subset $S$ of $D$. Then we say that $\{F_n\}$ {\it converges pointwise on $S$ to the limit function\/} $F$, defined by $$ F(x)=\lim_{n\to\infty}F_n(x),\quad x\in S. $$


A sequence $\{F_n\}$ of functions defined on a set $S$ {\it converges uniformly to the limit function $F$ on\/} $S$ if $$ \lim_{n\to\infty} || F_n-F\|_S=0. $$ Thus, $\{F_n\}$ converges uniformly to $F$ on $S$ if for each $\epsilon> 0$ there is an integer $N$ such that \begin{equation} \label{eq:4.4.1} \|F_n-F\|_S<\epsilon\mbox{\quad if\quad} n\ge N. \end{equation}


If $\{f_j\}^\infty_k$ is a sequence of real-valued functions defined on a set $D$ of reals, then $\sum_{j=k}^\infty  f_j$ is  an {\it infinite series\/} (or simply a {\it series\/}) of functions on $D$. The {\it partial sums of\/}, $\sum_{j=k}^\infty f_j$ are defined by $$ F_n=\sum^n_{j=k} f_j,\quad n\ge k. $$ If $\{F_n\}^\infty_k$ converges pointwise to a function $F$ on a subset $S$ of $D$, we say that $\sum_{j=k}^\infty f_j$ {\it converges pointwise to the sum $F$ on\/} $S$, and write $$ F=\sum_{j=k}^\infty f_j,\quad x\in S. $$ \newpage \noindent If $\{F_n\}$ converges uniformly to $F$ on $S$, we say that $\sum_{j=k}^\infty f_j$ {\it converges uniformly to $F$ on~$S$\/}.


An infinite series of the form \begin{equation}\label{eq:4.5.1} \sum^\infty_{n=0} a_n(x-x_0)^n, \end{equation} where $x_0$ and $a_0$, $a_1$, \dots, are constants, is called a {\it power series in $x-x_0$\/}. \bbox


The {\it vector sum\/} of $$ \mathbf{X}=(x_1,x_2, \dots,x_n)\mbox{\quad and\quad}\mathbf{Y}= (y_1,y_2, \dots,y_n) $$ is \begin{equation}\label{eq:5.1.1} \mathbf{X}+\mathbf{Y}=(x_1+y_1,x_2+y_2, \dots,x_n+y_n). \end{equation} If $a$ is a real number, the {\it scalar multiple of $\mathbf{X\/}$ by\/} $a$ is \begin{equation}\label{eq:5.1.2} a\mathbf{X}=(ax_1,ax_2, \dots,ax_n). \end{equation}


The {\it length\/} of the vector $\mathbf{X}=(x_1,x_2, \dots, x_n)$ is $$ |\mathbf{X}|=(x^2_1+x^2_2+\cdots+x^2_n)^{1/2}. $$ The {\it distance between points $\mathbf{X\/}$ and\/} $\mathbf{Y}$ is $|\mathbf{X}-\mathbf{Y}|$; in particular, $|\mathbf{X}|$ is the distance between $\mathbf{X}$ and the origin. If $|\mathbf{X}|=1$, then $\mathbf{X}$  is a {\it unit vector\/}. \bbox


The {\it inner product\/} $\mathbf{X}\cdot \mathbf{Y}$ of $\mathbf{X}=(x_1,x_2, \dots,x_n)$ and $\mathbf{Y}= (y_1,y_2, \dots,y_n)$ is $$ \mathbf{X}\cdot\mathbf{Y}=x_1y_1+x_2y_2+\cdots+x_ny_n. $$


$\mathbf{U}$ are in $\R^n$ and $\mathbf{U}\ne\mathbf{0}$. Then {\it the line through $\mathbf{X}_0$ in the direction of\/} $\mathbf{U}$ is the set of all points in $\R^n$ of the form $$ \mathbf{X}=\mathbf{X}_0+t\mathbf{U},\quad -\infty<t<\infty. $$ A set of points of the form $$ \mathbf{X}=\mathbf{X}_0+t\mathbf{U},\quad t_1\le t\le t_2, $$ is called a {\it line segment\/}. In particular, the line segment from $\mathbf{X}_0$ to $\mathbf{X}_1$ is the set of points of the form $$ \mathbf{X}=\mathbf{X}_0+t(\mathbf{X}_1-\mathbf{X}_0)=t\mathbf{X}_1+(1-t)\mathbf{X}_0, \quad 0\le t\le1. $$


If $\epsilon>0$, the {\it $\epsilon$-neighborhood of a point\/} $\mathbf{X}_{0}$ in $\R^n$ is the set $$ N_\epsilon(\mathbf{X}_0)|=\set{\mathbf{X}}{|\mathbf{X}-\mathbf{X}_0|<\epsilon}. \eqno{\bbox} $$


A sequence of points $\{\mathbf{X}_r\}$ in $\R^n$ {\it converges to the limit\/} $\overline{\mathbf{X}}$ if $$ \lim_{r\to\infty} |\mathbf{X}_r-\overline{\mathbf{X}}|=0. $$ In this case we write $$ \lim_{r\to\infty}\mathbf{X}_r=\overline{\mathbf{X}}. \eqno{\bbox} $$


If $S$ is  a nonempty subset of $\R^n$, then $$ d(S)=\sup\set{|\mathbf{X}-\mathbf{Y}|}{\mathbf{X},\mathbf{Y}\in S} $$ is the {\it diameter\/} of $S$. If $d(S)<\infty,$ $S$ is  {\it bounded\/}$;$ if $d(S)=\infty$, $S$ is {\it unbounded\/}.


A subset $S$ of $\R^n$ is  {\it connected\/} if it is impossible to represent $S$ as the union of two disjoint nonempty sets such that neither contains a limit point of the other; that is, if $S$ cannot be expressed as $S=A\cup B$, where \begin{equation}\label{eq:5.1.16} A\ne\emptyset,\quad B\ne\emptyset,\quad\overline{A}\cap B= \emptyset,\mbox{\quad and\quad} A\cap\overline{B}=\emptyset. \end{equation} If $S$ can be expressed in this way, then $S$ is {\it disconnected\/}.


A {\it region\/} $S$ in $\R^n$ is the union of an open connected set with some, all, or none of its boundary; thus, $S^0$ is connected, and every point of $S$ is a limit point of $S^0$.


We say that $f(\mathbf{X})$ {\it approaches the limit $L$ as $\mathbf{X\/}$ approaches\/} $\mathbf{X}_0$ and write $$ \lim_{\mathbf{X}\to\mathbf{X}_0} f(\mathbf{X})=L $$ if $\mathbf{X}_0$ is a limit point of $D_f$ and, for every $\epsilon>0$, there is a $\delta>0$ such that $$ |f(\mathbf{X})-L|<\epsilon $$ for all $\mathbf{X}$ in $D_f$ such that $$ 0<|\mathbf{X}-\mathbf{X}_0|<\delta. $$


We say that $f(\mathbf{X})$ {\it approaches $\infty$ as $\mathbf{X\/}$ approaches $\mathbf{X}_0$\/} and write $$ \lim_{\mathbf{X}\to\mathbf{X}_0} f(\mathbf{X})=\infty $$ if $\mathbf{X}_0$ is a limit point of $D_f$ and, for every real number $M$, there is a $\delta>0$ such that $$ f(\mathbf{X})>M\mbox{\quad whenever\quad} 0<|\mathbf{X}-\mathbf{X}_0|<\delta \mbox{\quad and\quad}\mathbf{X}\in D_f. $$ We say that \begin{eqnarray*} \lim_{\mathbf{X}\to\mathbf{X}_0} f(\mathbf{X})\ar=-\infty\\ \arraytext{if}\\ \lim_{{\mathbf{X}}\to\mathbf{X}_0} (-f)(\mathbf{X})\ar=\infty. \end{eqnarray*}


If $D_f$ is unbounded$,$ we say that $$ \lim_{|\mathbf{X}|\to\infty} f(\mathbf{X})=L\mbox{\quad (finite)\quad} $$ if for every $\epsilon>0$, there is a number $R$ such that $$ |f(\mathbf{X})-L|<\epsilon\mbox{\quad whenever\quad}\ |\mathbf{X}|\ge R \mbox{\quad and\quad}\mathbf{X}\in D_f. $$


If $\mathbf{X}_0$ is in $D_f$ and is a limit point of $D_f$, then we say that $f$ is {\it continuous at  $\mathbf{X\/}_0$\/}  if $$ \lim_{\mathbf{X}\to\mathbf{X}_0} f(\mathbf{X})=f(\mathbf{X}_0). \eqno{\bbox} $$


Let $\boldsymbol{\Phi}$ be a unit vector and $\mathbf{X}$ a point in $\R^n$.  {\it The directional derivative of $f$ at $\mathbf{X}$ in the direction of\/} $\boldsymbol{\Phi}$ is defined by $$ \frac{\partial f(\mathbf{X})}{\partial\boldsymbol{\Phi}}=\lim_{t\to 0}\frac {f(\mathbf{X}+ t\boldsymbol{\Phi})-f(\mathbf{X})}{ t} $$ if the limit exists. That is, $\partial f(\mathbf{X})/\partial\boldsymbol{\Phi}$ is the ordinary derivative of the function $$ h(t)=f(\mathbf{X}+t\boldsymbol{\Phi}) $$ at $t=0$, if $h'(0)$ exists.


A function $f$ is {\it differentiable\/} at $$  \mathbf{X}_0=(x_{10},x_{20}, \dots,x_{n0})) $$ if $\mathbf{X}_0\in D_f^0$ and there are  constants $m_1$, $m_2$, \dots$,$ $m_n$ such that \begin{equation}\label{eq:5.3.16} \lim_{\mathbf{X}\to\mathbf{X}_0} \frac{f(\mathbf{X})-f(\mathbf{X}_0)- \dst{\sum^n_{i=1}}\, m_i (x_i-x_{i0})}{ |\mathbf{X}-\mathbf{X}_0|}=0. \end{equation}


A vector-valued function  $\mathbf{G}=(g_1,g_2, \dots,g_n)$ is {\it differentiable\/} at $$ \mathbf{U}_0=(u_{10},u_{20}, \dots,u_{m0}) $$  if its component functions $g_1$, $g_2$, \dots, $g_n$ are differentiable at $\mathbf{U}_0$. \bbox


Suppose that  $r\ge1$ and all partial derivatives of $f$ of order $\le r-1$ are differentiable in a neighborhood of $\mathbf{X}_0$. Then the $r$th {\it differential of $f$ at\/} $\mathbf{X}_0$, denoted by $d^{(r)}_{\mathbf{X}_0}f$, is defined by \begin{equation} \label{eq:5.4.23} d^{(r)}_{\mathbf{X}_0}f=\sum_{i_1,i_2, \dots,i_r=1}^n \frac{\partial^rf(\mathbf{X}_0) }{\partial x_{i_r}\partial x_{i_{r-1}}\cdots\partial x_{i_1}} dx_{i_1}dx_{i_2}\cdots dx_{i_r}, \end{equation} where $dx_1$, $dx_2$, \dots, $dx_n$ are the differentials introduced in Section~5.3; that is, $dx_i$ is the function whose value at a point in $\R^n$ is the $i$th coordinate of the point. For convenience, we define $$ (d^{(0)}_{\mathbf{X}_0}f)=f(\mathbf{X}_0). $$ Notice that $d^{(1)}_{\mathbf{X}_0}f=d_{\mathbf{X}_0}f$. \bbox


A transformation $\mathbf{L}: \R^n \to \R^m$ defined on all of $\R^n$ is  {\it linear\/} if $$ \mathbf{L}(\mathbf{X}+\mathbf{Y})=\mathbf{L}(\mathbf{X})+\mathbf{L}(\mathbf{Y}) $$ for all $\mathbf{X}$ and $\mathbf{Y}$ in $\R^n$ and $$ \mathbf{L}(a\mathbf{X})=a\mathbf{L}(\mathbf{X}) $$ for all $\mathbf{X}$ in $\R^n$ and real numbers $a$.


\begin{alist} \item % (a)  If $c$ is a real number and $\mathbf{A}=[a_{ij}]$ is an $m\times n$ matrix, then $c\mathbf{A}$ is the $m\times n$ matrix defined by $$ c\mathbf{A}=[ca_{ij}]; $$ that is, $c\mathbf{A}$ is obtained by multiplying every entry of $\mathbf{A}$ by $c$. \item % (b) If $\mathbf{A}=[a_{ij}]$ and $\mathbf{B}=[b_{ij}]$ are $m\times n$ matrices, then the {\it sum\/}  $\mathbf{A}+  \mathbf{B}$  is the $m\times n$ matrix $$ \mathbf{A}+\mathbf{B}=[a_{ij}+b_{ij}]; $$ that is, the sum of two $m\times n$ matrices is obtained by adding corresponding entries. The sum of two matrices is not defined unless they have the same number of rows and the same number of columns. \item % (c) If $\mathbf{A}=[a_{ij}]$ is an $m\times p$ matrix and $\mathbf{B}= [b_{ij}]$ is a $p\times n$ matrix, then the {\it product\/} $\mathbf{C}=\mathbf{AB}$ is the $m\times n$ matrix with $$ c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ip}b_{pj}=\sum^p_{k=1} a_{ik}b_{kj},\quad 1\le i\le m,\ 1\le j\le n. $$ Thus, the $(i,j)$th entry of $\mathbf{AB}$ is obtained by multiplying each entry in the $i$th row of $\mathbf{A}$ by the corresponding entry in the $j$th column of $\mathbf{B}$ and adding the products. This definition requires that $\mathbf{A}$ have the same number of columns as $\mathbf{B}$ has rows. Otherwise, $\mathbf{AB}$ is undefined. \end{alist}


The {\it norm\/}$,$ $\|\mathbf{A}\|,$ of an $m\times n$ matrix $\mathbf{A}=[a_{ij}]$ is the smallest number such that $$ |\mathbf{AX}|\le\|\mathbf{A}\|\,|\mathbf{X}| $$ for all $\mathbf{X}$ in $\R^n.$ \bbox


Let $\mathbf{A}=[a_{ij}]$ be an $n\times n$ matrix$,$ with $n\ge2.$ The {\it cofactor\/} of an entry $a_{ij}$ is $$ c_{ij}=(-1)^{i+j}\det(\mathbf{A}_{ij}), $$ where $\mathbf{A}_{ij}$ is the $(n-1)\times(n-1)$ matrix obtained by deleting  the $i$th row and $j$th column of $\mathbf{A}.$ The {\it adjoint\/} of $\mathbf{A},$ denoted by $\adj(\mathbf{A}),$ is the $n\times n$ matrix whose $(i,j)$th entry is $c_{ji}.$


A transformation $\mathbf{F}: \R^n\to \R^n$ is {\it regular\/} on an open set $S$ if $\mathbf{F}$ is one-to-one and continuously differentiable on $S$, and $J\mathbf{F}(\mathbf{X})\ne0$ if $\mathbf{X}\in S$. We will also say that $\mathbf{F}$  is regular on an arbitrary  set $S$ if $\mathbf{F}$ is regular on an open set containing $S$.


A {\it coordinate rectangle\/} $R$ in $\R^n$ is the Cartesian product of $n$ closed intervals; that is, $$ R=[a_1,b_1]\times [a_2,b_2]\times\cdots\times [a_n,b_n]. $$ The {\it content\/} of $R$ is $$ V(R)=(b_1-a_1)(b_2-a_2)\cdots (b_n-a_n). $$ The numbers $b_1-a_1$, $b_2-a_2$, \dots, $b_n-a_n$ are  the {\it edge lengths\/} of $R$. If they are equal, then $R$ is a {\it coordinate cube\/}.  If $a_r=b_r$ for some $r$, then $V(R)=0$ and we say that $R$ is {\it degenerate\/}; otherwise, $R$ is {\it nondegenerate\/}.  \bbox


Let $f$ be a real-valued function defined on a  rectangle $R$ in $\R^n$. We say that  $f$ is {\it Riemann integrable on\/} $R$  if there is a number $L$ with the following property: For every $\epsilon>0$, there is a $\delta>0$ such that $$ \left|\sigma-L\right|<\epsilon $$ if $\sigma$ is any Riemann sum of $f$ over a partition ${\bf P}$ of $R$ such that $\|{\bf P}\|<\delta$. In this case, we say that  $L$ is the  {\it Riemann integral of $f$ over\/} $R$, and write $$ \int_R f(\mathbf{X})\,d\mathbf{X}=L. \eqno{\bbox} $$


If $f$ is bounded on a rectangle $R$ in $\R^n$ and ${\bf P}=\{R_1,R_2, \dots,R_k\}$ is a partition of $R$, let $$ M_j=\sup_{\mathbf{X}\in R_j}f(\mathbf{X}),\quad m_j= \inf_{\mathbf{X}\in R_j}f(\mathbf{X}). $$ The {\it upper sum\/} of $f$ over ${\bf P}$ is $$ S({\bf P})=\sum_{j=1}^k M_jV(R_j), $$ and the {\it upper integral  of $f$ over\/} $R$, denoted by $$ \overline{\int_R}\,f(\mathbf{X})\,d\mathbf{X}, $$  is the infimum of all  upper sums. The {\it lower sum of $f$ over\/} ${\bf P}$ is $$ s({\bf P})=\sum_{j=1}^k m_jV(R_j), $$ and the {\it lower integral  of $f$ over \/}$R$, denoted by $$ \underline{\int_R}\, f(\mathbf{X})\,d\mathbf{X}, $$  is the supremum of all  lower sums. \bbox


A subset $E$ of $\R^n$ has zero content if for each $\epsilon>0$ there is a finite set of rectangles $T_1$, $T_2$, \dots, $T_m$ such that \begin{equation}\label{eq:7.1.24} E\subset\bigcup_{j=1}^m T_j \end{equation} and \begin{equation}\label{eq:7.1.25} \sum_{j=1}^m V(T_j)<\epsilon. \end{equation}


Suppose that $f$ is bounded on a bounded subset of $S$ of $\R^n$, and let \begin{equation}\label{eq:7.1.36} f_S(\mathbf{X})=\left\{\casespace\begin{array}{ll} f(\mathbf{X}),&\mathbf{X}\in S,\\[2\jot]  0,&\mathbf{X}\not\in S.\end{array}\right. \end{equation} Let $R$ be a rectangle containing $S$. Then {\it the integral of $f$ over $S$\/} is defined to be $$ \int_S f(\mathbf{X})\,d\mathbf{X}=\int_R f_S(\mathbf{X})\,d\mathbf{X} $$ if  $\int_R f_S(\mathbf{X})\, d\mathbf{X}$ exists. \bbox


If $S$ is a bounded subset of $\R^n$ and the integral $\int_S\,d\mathbf{X}$ (with integrand  $f\equiv1$) exists, we call $\int_S\,d\mathbf{X}$ the {\it content\/} (also, {\it area\/} if $n=2$ or {\it volume\/} if $n=3$) of $S$, and denote it by $V(S)$; thus, $$ V(S)=\int_S\,d\mathbf{X}. $$


A {\it differentiable surface\/} $S$ in $\R^n\ (n>1)$ is the image of a compact subset $D$ of $\R^m$, where $m< n$, under a continuously differentiable transformation $\mathbf{G}: \R^m\to \R^n$. If $m=1$, $S$ is also called a {\it differentiable curve\/}.


If $\mathbf{A}=[a_{ij}]$ is an $n \times n$ matrix$,$ then $$ \max\set{\sum_{j=1}^n |a_{ij}|}{1\le i\le n} $$ is the {\it infinity norm of\/} $A,$  denoted by $\|A\|_\infty$.


A {\it metric space\/} is a nonempty set $A$ together with a real-valued function $\rho$ defined on $A\times A$ such that  if $u$, $v$, and $w$ are arbitrary members of $A$, then \begin{alist} \item % (a) $\rho(u,v)\ge 0$, with equality if and only if $u=v$; \item % (b) $\rho(u,v)=\rho(v,u)$; \item % (c) $\rho(u,v)\le\rho(u,w)+\rho(w,v)$. \end{alist} We say that $\rho$ is a {\it metric\/} on $A$. \bbox


A {\it vector space\/} $A$ is a nonempty set of elements called {\it vectors\/} on which two operations,  vector addition and scalar multiplication (multiplication by real numbers) are defined, such that the following assertions are true for all $\mathbf{U}$, $\mathbf{V}$, and $\mathbf{W}$ in $A$ and all real numbers $r$ and $s$:\\ \phantom{1}1.  $\mathbf{U}+\mathbf{V}\in A$;\\ \phantom{1}2. $\mathbf{U}+\mathbf{V}=\mathbf{V}+\mathbf{U}$;\\ \phantom{1}3.  $\mathbf{U}+(\mathbf{V}+\mathbf{W})=(\mathbf{U}+\mathbf{V})+\mathbf{W}$;\\ \phantom{1}4. There is a vector $\mathbf{0}$ in $A$ such that $\mathbf{U}+\mathbf{0}=\mathbf{U}$;\\ \phantom{1}5. There is a vector $-\mathbf{U}$ in $A$ such that $\mathbf{U}+(-\mathbf{U})=\mathbf{0}$;\\ \phantom{1}6. $r\mathbf{U}\in A$;\\ \phantom{1}7. $r(\mathbf{U}+\mathbf{V})=r\mathbf{U}+r\mathbf{V}$;\\ \phantom{1}8. $(r+s)\mathbf{U}=r\mathbf{U}+s\mathbf{U}$;\\ \phantom{1}9. $r(s\mathbf{U})=(rs)\mathbf{U}$; \\ 10. $1\mathbf{U}=\mathbf{U}$. \bbox


A {\it normed vector space\/}  is a vector space $A$ together with a real-valued function $N$  defined on $A$, such that  if $u$ and $v$ are arbitrary vectors in $A$ and $a$ is a real number, then \begin{alist} \item % (a) $N(u)\ge 0$ with equality if and only if $u=0$; \item % (b) $N(au)=|a|N(u)$; \item % (c) $N(u+v)\le N(u)+N(v)$. \end{alist} We say that $N$ is a {\it norm\/} on $A$, and $(A,N)$ is a {\it normed vector space\/}.


If $p\ge 1$ and $\mathbf{X}=(x_1,x_2, \dots,x_n)$, let \begin{equation} \label{eq:8.1.3} \|\mathbf{X}\|_p =\left(\sum_{i=1}^n|x_i|^p\right)^{1/p}. \end{equation} The metric induced on $\R^n$ by this norm is $$ \rho_p(\mathbf{X},\mathbf{Y}) =\left(\sum_{i=1}^n|x_i-y_i|^p\right)^{1/p}. \eqno{\bbox} $$


If $u_0\in A$  and $\epsilon>0$, the set $$ N_\epsilon(u_0)=\set{u\in A}{\rho(u_0,u)<\epsilon} $$ is called an {\it $\epsilon$-neighborhood\/} of $u_0$. (Sometimes we call $S_\epsilon$ the {\it open ball of radius $\epsilon$ centered at $u_0$\/}.) If a subset $S$ of $A$ contains an $\epsilon$-neighborhood of $u_0$, then $S$ is a {\it neighborhood\/} of $u_0$, and $u_0$ is an {\it interior point\/} of $S$. The set of interior points of $S$ is the {\it interior\/} of $S$, denoted by $S^0$. If every point of $S$ is an interior point (that is, $S^0=S$), then $S$ is {\it open\/}. A set $S$ is  {\it closed\/} if $S^c$ is open.


Then \begin{alist} \item % (a) $u_0$ is a {\it limit point\/} of $S$ if every deleted neighborhood of $u_0$ contains a point of~$S$. \item % (b) $u_0$ is a {\it boundary point\/} of $S$ if every neighborhood of $u_0$ contains at least one point in $S$ and one not in $S$. The set of boundary points of $S$ is the {\it boundary\/} of $S$, denoted by $\partial S$. The {\it closure\/} of $S$, denoted by $\overline{S}$, is defined by $\overline{S}=S\cup \partial S$. \item % (c) $u_0$ is an {\it isolated point\/} of $S$ if $u_0\in S$ and there is a neighborhood of $u_0$ that contains no other point of $S$. \item % (d) $u_0$ is {\it exterior } to $S$ if $u_0$ is in the interior of $S^c$. The collection of such points is the {\it exterior\/} of $S$. \bbox \end{alist}


A sequence $\{u_n\}$ in a metric space $(A,\rho)$  {\it converges\/} to $u\in A$ if \begin{equation} \label{eq:8.1.16} \lim_{n\to\infty}\rho(u_n,u)=0. \end{equation} In this case we say that $\lim_{n\to\infty}u_n=u$. \bbox


A sequence $\{u_n\}$ in a metric space $(A,\rho)$ is  a {\it Cauchy sequence\/}  if for every $\epsilon>0$ there is an integer $N$ such that \begin{equation} \label{eq:8.1.17} \rho(u_n,u_m)<\epsilon\mbox{\quad and \quad}m,n>N. \end{equation}


A metric space $(A,\rho)$ is {\it complete\/}  if every Cauchy sequence in $A$ has a limit.


If $\rho$ and $\sigma$ are both metrics on a set $A$, then $\rho$ and $\sigma$ are {\it equivalent \/} \hskip-.2em if there are positive constants  $\alpha$  and $\beta$ such that \begin{equation} \label{eq:8.1.18} \alpha\le\frac{\rho(x,y)}{\sigma(x,y)}\le\beta \mbox{\quad for all \quad}x,y\in A\mbox{\quad such that \quad}x\ne y. \end{equation}


The {\it diameter\/} of a nonempty subset $S$ of $A$ is $$ d(S)=\sup\set{\rho(u,v)}{u,\, v\in T}. $$ If $d(S)<\infty$ then $S$ is {\it bounded\/}. \bbox


A set $T$  is {\it compact\/} if it has the Heine--Borel property.


A  set $T$  is {\it totally bounded\/}  if for every $\epsilon>0$ there is a finite set $T_\epsilon$ with the following property: if $t\in T$, there is an $s\in T_\epsilon$ such that $\rho(s,t)<\epsilon$. We say that $T_\epsilon$ is a {\it finite $\epsilon$-net for $T$\/}. \bbox


A subset $T$ of $C[a,b]$ is {\it uniformly bounded\/} if there is a constant $M$ such that \begin{equation} \label{eq:8.2.6} |f(x)|\le M \mbox{\quad if \quad} a\le x\le b\mbox{\quad and \quad} f\in T. \end{equation} A subset $T$ of $C[a,b]$ is {\it equicontinuous\/} if for each $\epsilon>0$ there is a $\delta>0$ such that \begin{equation} \label{eq:8.2.7} |f(x_1)-f(x_2)|\le \epsilon \mbox{\quad if \quad} x_1,x_2\in [a,b],\quad |x_1-x_2|<\delta,\mbox{\quad and \quad}f\in T. \end{equation}


We say that $$ \lim_{u\to \widehat u}f(u)=\widehat v $$ if $\widehat u\in\overline D_f$ and for each $\epsilon>0$ there is a $\delta>0$  such that \begin{equation} \label{eq:8.3.1} \sigma(f(u),\widehat v)<\epsilon\mbox{\quad if \quad} u\in D_f \mbox{\quad and \quad} 0<\rho(u,\widehat u)<\delta. \end{equation}


We say that $f$ is {\it continuous\/} at $\widehat u$ if $\widehat u\in D_f$ and for each $\epsilon>0$ there is a $\delta>0$  such that \begin{equation} \label{eq:8.3.2} \sigma(f(u),f(\widehat u))<\epsilon\mbox{\quad if \quad} u\in D_f\cap N_\delta(\widehat u). \end{equation} If $f$ is  continuous at every point of a set $S$, then $f$ is {\it continuous on\/} S. \bbox


A function $f$ is {\it uniformly continuous\/} on a subset $S$ of $D_f$ if for each $\epsilon>0$ there is a $\delta>0$ such that $$ \sigma(f(u),f(v))<\epsilon\mbox{\quad whenever \quad} \rho(u,v)<\delta\mbox{\quad and \quad}u,v\in S. $$


If $f:(A,\rho)\to (A,\rho)$  is defined on all of $A$ and there is a constant $\alpha$ in $(0,1)$ such that \begin{equation} \label{eq:8.3.7} \rho(f(u),f(v))\le\alpha\rho(u,v) \mbox{\quad for all\quad} (u,v)\in A\times A, \end{equation} then $f$ is  a {\it contraction\/} of $(A,\rho)$. \bbox