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\begin{document}






\iftitle{Review of Basic Analysis\\{\normalsize Integration Workshop 2018, Department of Mathematics, University of Arizona}}
\vspace{-3ex}\ifauthor{Lecture notes by Ibrahim Fatkullin}{ibrahim@math.arizona.edu}


	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Sequences and Series}

\Def A set $\set M$ and a function $d: \set M\times \set M\to\R^+$ are called a {\bf metric space} if
%
\begin{enumerate}
	\item $d(x,y)=d(y,x)$
	\item $d(x,y)=0$ \quad {\em iff} \quad  $x=y$
	\item $d(x,y)\leq d(x,z)+ d(z,y)$ \qquad {\em (triangle inequality)}
\end{enumerate}


\Def A sequence $\{x_n\}$ {\bf converges} to $x$, if for every $\epsilon>0$ there exists $N$, such that for all $n\geq N$, $$d(x,x_n)\leq\epsilon.$$

\Def A sequence $\{x_n\}$ is called {\bf Cauchy} (or {\bf fundamental}) if for every $\epsilon>0$ there exists $N$, such that $$d(x_m,x_n)\leq\epsilon,\qquad\text{for all}\qquad m,n\geq N.$$

\Def A metric space is called {\bf complete} if every Cauchy sequence converges.

\Def A metric space is called {\bf compact} if any sequence has a converging subsequence.

\Def Series
%
\begin{equation}\label{eq:series}
	\sum_{n=1}^\infty x_n
\end{equation}
%
{\bf converges} if its partial sums, $\displaystyle S_N=\sum_{n=1}^N x_N$, converge as $N\to\infty$.

\Def Series \eqref{eq:series} {\bf converges absolutely} if $$\lim_{N\to\infty}\sum_{n=1}^N|x_n|<\infty.$$

\Def Given a power series, $\displaystyle\sum_{n=0}^\infty c_nz^n$, define $$\alpha=\limsup_{n\to\infty}\sqrt[n]{|c_n|};\qquad R=\frac{1}{\alpha}.$$ $R$ is called the {\bf radius of convergence} of the series, the latter converges if $|z|<R$ and diverges if $|z|>R$.




\newpage 
\subsection{Convergence Tests}

\paragraph{Root test.} Let $\alpha=\limsup_{n\to\infty}\sqrt[n]{|x_n|}$. Series \eqref{eq:series} converges (absolutely) or diverges if $\alpha<1$ or $\alpha>1$ respectively. (More analysis is required if $\alpha=1$)
%

\paragraph{Ratio test.} Series \eqref{eq:series} converges (absolutely) if  $\limsup_{n\to\infty}|x_{n+1}/x_n|<1$ and diverges if there exists some number $N$ such that $|x_{n+1}/x_n|\geq 1$ for all $n>N$.
%

\paragraph{Comparison test.} If $\displaystyle\lim_{n\to\infty}\left|\frac{x_n}{y_n}\right|=C\in(0,\infty)$, series $\displaystyle\sum_{n=1}^\infty x_n$ converges absolutely iff series $\displaystyle\sum_{n=1}^\infty y_n$ does.


\subsection{Is there a ``boundary'' between converging and diverging series?}

The series  $$\sum_{n=1}^\infty\frac{1}{n^\alpha}$$ converges for $\alpha>1$ and diverges for $\alpha\leq1$. Thus the exponent $\alpha=1$ corresponds to the ``boundary'' for power-law decay rates between converging and diverging series. However, for more general functions, how ``close'' can we get to $1/n$ while still maintaining convergence? For example, $$\sum_{n=2}^\infty\frac{1}{n(\ln n)^\alpha}$$ converges for all $\alpha>1$ and diverges for $\alpha\leq 1$. So we lifted our boundary a bit, from $1/n$ to $1/(n\ln n)$. We can go even further and observe that $$\sum_{n=3}^\infty\frac{1}{n\ln n(\ln\ln n)^\alpha}$$ converges for all $\alpha>1$ and diverges for $\alpha\leq 1$. Etc, etc: we can keep adding more iterated logarithms (or other functions) in a similar manner. Is there some limit to this process? In other words, e.g, is there some special monotone-decreasing sequence $\{b_n\}$ such that whenever $c_n/b_n\to0$ (as $n\to\infty$) the series $\sum c_n$ converges and whenever $b_n/d_n\to0$, the series $\sum d_n$ diverges? 



\subsection{Fun Stuff}

Consider the geometric series, $$\sum_{n=0}^\infty z^n=\frac{1}{1-z},\qquad |z|<1.$$ Pretending that this formula is valid for arbitrary $z\neq 1$, we can ``derive'' that $1-1+1-1+\cdots=1/2$, or $1+2+4+8+\cdots=-1$. In this case the divergent sum acquires meaning via analytic continuation of some appropriately chosen function outside of the radius of convergence of its power series. In a similar fashion one can get such formulas as, e.g.,
%
$$
	1-2+3-4+\cdots\;\coloneq\;\frac{1}{(1+z)^2}\;\Bigg|_{z=1}=\frac{1}{4};\qquad\qquad
	1+2+3+4+\cdots\;\coloneq\;\zeta(-1)=-\frac{1}{12}.
$$





\section{Continuity and differentiation}
				
Unless specified otherwise, we consider functions between metric spaces $\set X$ and $\set Y$.				
				
\Def A function $f$ is called {\bf continuous at $\boldsymbol{x_0}$} if for every $\epsilon>0$ there exists $\delta>0$, such that for all $x\in\set X$ with $d_{\set X}(x,x_0)<\delta$, $d_{\set Y}\big(f(x),f(x_0)\big)<\epsilon$. A function which is continuous at every point of $\set X$ is called {\bf continuous in $\boldsymbol{\set X}$.}
		
\Def A function $f$ is called {\bf uniformly continuous} if for every $\epsilon>0$ there exists $\delta>0$, such that for all $x_1,~x_2\in\set X$ with $d_{\set X}(x_1,x_2)<\delta$, $d_{\set Y}\big(f(x_1),f(x_2)\big)<\epsilon$.
		
\vspace{2em}\noindent Assume that our metric spaces are also {\em normed} linear vector spaces with metric and norm related via $d(f,g)=\|f-g\|$.
		
\Def Suppose $\set O$ is an open set in $\set X$; $f$ maps $\set O$ into $\set Y$; $x_0\in\set O$. If there exists a {\em bounded} linear operator $\D f(x_0)$, such that
%
\begin{equation}\label{eq:derive}
	\lim_{\|x\|_{\set X}\to0}\frac{\|f(x_0+x)-f(x_0)-\D f(x_0)x\|_{\set Y}}{\|x\|_{\set X}}=0,
\end{equation}
%
then $f$ is called {\bf differentiable at $\boldsymbol{x_0}$}, and $\D f(x_0)$ is called the {\bf (Fr\'echet) derivative} or {\bf differential} of $f$ at $x_0$. If $f$ is differentiable at every point in $\set O$, we call $f$ differentiable in $\set O$. The {\em determinant} of the operator $\D f(x_0)$ (if well-defined) is called the {\bf Jacobian} of $f$ at $x_0$.	

\subsection{Some Important Results}		
		
\paragraph{Mean value theorem.} Suppose $f$ is continuous on $[a,b]$ and differentiable in $(a,b)$. There exists \mbox{$x\in(a,b)$}, such that $$f^\prime(x)=\frac{f(a)-f(b)}{a-b}.$$
		
\paragraph{Taylor's theorem (1d).} Suppose $f\in\spc{C}^{n-1}[a,b]$ and $f^{(n)}(x)$ exists for all $x\in(a,b)$. For all $x$ and $y$ such that $a<x<y<b$, there exists $\xi\in[x,y]$ such that $$f(x)=\sum_{k=0}^{n-1}\frac{f^{(k)}(y)}{k!}(x-y)^k\,+\,\frac{f^{(n)}(\xi)}{n!}(x-y)^n.$$
		
\paragraph{Taylor's theorem (multi-d).} Let $\bar{\set B}$ be a closed ball centered at the origin in $\R^n$; $f\in\spc C^{n}(\bar{\set B})$; $x\in{\set B}$. Then $$f(x)=\sum_{|\vv\alpha|<n}\frac{x^{\vv\alpha}}{\vv\alpha!}\;\pd^{\vv\alpha}\!f(0)\,+\,\sum_{|\vv\alpha|=n}\frac{x^{\vv\alpha}}{\vv\alpha!}\;\pd^{\vv\alpha}\!f(\xi x),\qquad\text{for some\qquad}\xi\in[0,1].$$

\paragraph{Inverse function theorem.} Assume that $f$ is a continuously differentiable function from $\R^n$ and $\D f(x)$ is invertible. Then $f$ is invertible in some neighborhood of $x$ and its inverse is continuously differentiable in some neighborhood of $f(x)$.

\paragraph{Implicit function theorem.} Assume that $F$ is a continuously differentiable function from (an open subset) $\set O\subset\R^n\times\R^m$ into $\R^m$; $(x,y)\in\set O$; $F(x,y)=0$; and $\D F$ is one-to-one. Then there exists a neighborhood $\set N\subset\R^n$ containing $x$ and a function $f:\set N\to\R^m$, such that $f(x)=y$ and $F\big(x,f(x)\big)=0$ for all $x\in\set N$.




\newpage
\section{Integration, theorems relating integrals and derivatives}

\Def A finite ordered subset of $[a,b]$, $\vv\pi=(\pi_1,\ldots,\pi_n)$, such that $$a=\pi_1<\pi_2<\ldots<\pi_{n-1}<\pi_n=b$$ is called a {\bf partition} of $[a,b]$. We say $\vv\pi_2$ is a {\bf refinement} of $\vv\pi_1$ if $\vv\pi_1\subset\vv\pi_2$. A sequence of partitions $\{\vv\pi^n\}$ is called {\bf fine} if each partition in the sequence is a refinement of the previous one and $$\lim_{n\to\infty}\max_{k=2,\dots,|\vv\pi^n|}(\pi^n_{k}-\pi^n_{k-1})=0.$$

\Def Suppose the functions $f$ and $g$ are such that following limit exists and is the same for all fine sequences of partitions of $[a,b]$ and all $\vv x(\vv\pi)=(x_2,\ldots,x_{|\vv\pi|})$ such that $x_k\in[\pi_{k-1},\pi_{k}]$, $k=2,\ldots,|\vv\pi|$: $$\lim_{n\to\infty}\sum_{k=2}^{|\vv\pi^n|}f\big(x_k(\vv\pi^n)\big)\big|g(\pi^n_k)-g(\pi^n_{k-1})\big|.$$ It is then called the {\bf Riemann-Stieltjes integral}  of $f$ with respect to $g$ over $[a,b]\ssa\Omega$  and is denoted by
%
\begin{equation}\label{eq:int}
	\int_a^b f(x)\md g(x)\qquad\text{or}\qquad\int_\Omega f\md g.
\end{equation}

\bul If $g$ is differentiable, then Riemann-Stieltjes integral can be related to the usual Riemann integral, $$\int_\Omega f\md g = \int_\Omega f(x) g^\prime(x)\md x.$$

\Def A function $f:[a,b]\to\R$ is called of {\bf bounded variation} if $$\V_a^b(f)\coloneq\sup_{\vv\pi\in\set P[a,b]}\sum_{n=2}^{|\vv\pi|}|f(\pi_n)-f(\pi_{n-1})|<\infty.$$ Here $\set P[a,b]$ denotes the set of all partitions of $[a,b]$. The space of all functions of bounded variation on $[a,b]$ is denoted by $\BV[a,b]$.





\subsection{Some Important Results}



\paragraph{Existence of Riemann-Stieltjes integral} Suppose $f\in\spc C[a,b]$ and $g\in\BV[a,b]$, then the Riemann-Stieltjes integral \eqref{eq:int} exists. 

\bul For a given $g\in\BV[a,b]$, the class of functions integrable with respect to $g$ is larger than $C[a,b]$ and essentially includes all Riemann-integrable functions which do not share points of discontinuity with $g$.



\paragraph{Fundamental theorem of calculus.} Let $f\in\spc C[a,b]$, $g\in\BV[a,b]$, then $$\int_a^b\md g=g(b)-g(a);\quad\text{if in addition~}g\in\spc C[a,b],\quad\text{then}\quad\frac{\md}{\md g}\int_a^xf(y)\md g(y)=f(x)\quad\text{for all~}x\in[a,b].$$
%
Here $\displaystyle\frac{\md F(x)}{\md g(x)}\coloneq\lim_{\epsilon\to0}\frac{F(x+\epsilon)-F(x)}{g(x+\epsilon)-g(x)}$ --- (essentially) the {\bf Radon-Nikodym derivative} of $F$ with respect to $g$.

\paragraph{Change of variables.} Suppose $g,h\in\BV(\Omega)$; $f,\md g/\md h\in\spc C[a,b]$, then $$\int_\Omega f\md g=\int_\Omega f\frac{\md g}{\md h}\md h.$$

\paragraph{Integration by parts.} Suppose $f,g\in\BV[a,b]$, $f\in\spc C[a,b]$, then $$\int_a^b f\md g=f(b)g(b)-f(a)g(a)-\int_a^b g\md f.$$

\paragraph{Integral mean value theorem I.} Let $f$ be continuous and $g$ monotone on $[a,b]$, then there exists $x\in[a,b]$, such that $$\int_a^bf\md g = f(x)\big[g(b)-g(a)\big].$$

\paragraph{Integral mean value theorem II.} Let $f$ be monotone and $g$ be continuous on $[a,b]$, then there exists $x\in[a,b]$, such that $$\int_a^bf\md g = f(a)\big[g(x)-g(a)\big]+f(b)\big[g(b)-g(x)\big].$$
 



\section{Sequences of Functions}

\Def A sequence of functions $\{f_n\}$ converges to $f$ {\bf point-wise} in $\set X$ if for every $x\in\set X$, $$\lim_{n\to\infty}f_n(x)=f(x).$$

\Def A sequence of functions $\{f_n\}$ converges to $f$ {\bf uniformly} in $\set X$ if for every $\epsilon>0$ there exists $N$ such that for all $n>N$ and all $x\in\set X$, $$d\big(f_n(x),f(x)\big)<\epsilon.$$

\Def A family of functions, $\set F$, is called {\bf equicontinuous} if for all $\epsilon>0$ there exists $\delta>0$, such that whenever $d_{\set X}(x_1,x_2)<\delta$, $$d_{\set Y}\big(f(x_1),f(x_2)\big)<\epsilon\quad\text{ for all }\quad f\in\set F.$$




\subsection{Some Important Results}




\paragraph{Weierstrass M-test.} If $\sup_{x\in\set X}|f_n(x)|<M_n$ and the series $\sum M_n$ converges, then $\sum f_n(x)$ converges uniformly in $\set X$.

\paragraph{Uniform convergence theorem.} A uniform limit of continuous functions is continuous.

\paragraph{Monotone convergence theorem.} A point-wise monotone sequence of continuous functions converging to a continuous function on a compact set does so uniformly.

\paragraph{Exchanging the order of limits and integration.} Suppose $f_n$ converge uniformly to $f$ in $\Omega$ and each $f_n$ is integrable with respect to $g$ over $\Omega$, then $$\lim_{n\to\infty}\int_\Omega f_n\md g=\int_\Omega f\md g.$$

\paragraph{Exchanging the order of limits and differentiation.} Suppose $f^\prime_n$ converge uniformly on $[a,b]$ and  $f_n$ converge at some $x_0\in[a,b]$, then $f_n$ converge uniformly on $[a,b]$ to some differentiable function $f$ and$$\lim_{n\to\infty} f^\prime_n(x)=f^\prime(x).$$

\paragraph{Stone-Weierstrass theorem.} Continuous functions on $\R^n$ may be uniformly approximated by polynomials on compact subsets of $\R^n$.

\paragraph{Arzel\`a-Ascoli Theorem.} Every infinite equicontinuous family of maps between compact metric spaces contains a uniformly converging sequence.


	

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\subsection{Multidimensional Integrals}

\paragraph{Exchanging the order of integration.}
\paragraph{Change of variables.}
\paragraph{Stokes' theorem.} $$\int_\Omega\md\omega=\int_{\pd\Omega}\omega.$$

















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