Overview §

Columbia University, Fall 2024 – S. Hirsch

Course description

Real numbers, metric spaces, elements of general topology, sequences and series, continuity, differentiation, integration, uniform convergence, Ascoli-Arzela theorem, Stone-Weierstrass theorem.

Study status §

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Topics §

The real and complex numbers §

Definition: Real numbers

The field of real numbers is a tuple $(\mathbb{R},+,\cdot ,\le )$ consisting of a set $\mathbb{R}$, the maps $+,\cdot :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined by

$$+(x,y):=x+y\cdot (x,y)=xy$$

respectively, and a order relation $\le$ satisfying the following:

• (A1), (M1) Existence of a neutral element. See (F4) Identities.
• (A2), (M2) Existence of inverses. See (F5) Additive inverses and (F6) Multiplicative inverses.
• (A3), (M3) Associativity. See (F1) Associativity.
• (A4), (M4) Commutativity. See (F2) Commutativity.
• (D) Distributivity of addition over multiplication. For all $x,y,z\in \mathbb{R}$, we have $(x+y)\cdot z=x\cdot z+y\cdot z$.

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Basic topology §

Metric spaces §

Definition: Metric, metric space

A metric on a set $X$ is a function $d:X\times X\to \mathbb{R}$ that satisfies the following conditions for all $x,y,z\in X$:

• (M1) Non-negativity and identity. $d(x,y)\ge 0$, with $d(x,y)=0$ if and only if $x=y$;
• (M2) Symmetry. $d(x,y)=d(y,x)$;
• (M3) Triangle inequality. We have $d(x,y)\le d(x,z)+d(z,y)$.

The value $d(x,y)$ is often called the distance between $x$ and $y$ in the metric $d$. A metric space is a pair $(X,d)$ where $X$ is any set and $d$ is a metric on $X$.

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Definition (Rudin 2.18): Elements and subsets of metric spaces

Let $(X,d)$ be a metric space.

• A neighborhood of a point $p\in X$ is the set

$$B_{ϵ}(p)=\{q\in X|d(p,q)<r\}$$

for some radius $r>0$.

• A point $p$ is a limit point of the set $E\subseteq X$ if every neighborhood of $p$ contains a point $q\in E$ where $q\ne p$; that is, every neighborhood of $p$ intersects $E$ at some point other than itself.
• A point $p$ is an interior point of $E$ if there exists a neighborhood $U$ of $p$ such that $U\subseteq E$.
• $E$ is closed if every limit point of $E$ is a point of $E$, and open if every point of $E$ is an interior point of $E$.
• $E$ is dense in $X$ if every point of $X$ is either a limit point of $E$, in $E$, or both.

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Compactness §

Definition: Open covers in metric spaces

Let $(X,d)$ be a metric space and $A\subseteq Z$. A family of sets $\{G_{\alpha }{\}}_{\alpha \in J}$ is called an open cover of $A$ if all $G_{\alpha }$ are open and if $A\subseteq {\bigcup }_{\alpha \in J}{G}_{\alpha }$.

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Definition: Compact sets in metric spaces

Let $(X,d)$ be a metric space. We say that $A\subseteq X$ is compact if every open cover ${\bigcup }_{\alpha \in J}{G}_{\alpha }$ of $A$ has a finite subcover. That is, there exists some other indexing set $K$ such that $J\subseteq K$ and $|K|<\infty$.

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Definition: Bounded sets in metric spaces

Let $(X,d)$ be a metric space with a subset $A\subset X$. Then $A$ is bounded if there exists a point $x\in X$ and some $r>0$ such that $A$ is contained in the $r$-ball about $x$ given by

$$B(x,r)=\{y\in X|d(x,y)<r\}.$$

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Lemma: Hausdorff property for metric spaces

Let $(X,d)$ be a metric space. Given $x,y\in X$ where $x\ne y$, there exist open sets $A,B$ such that $x\in A$ and $y\in B$ and $A\cap B=\varnothing$.

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Proposition (Rudin 2.34, 2.35): Compact sets are closed and bounded; closed subsets are also compact

Let $(X,d)$ be a metric space and let $A\subset X$ be compact. Then:

• (i) $A$ is bounded;
• (ii) If $B\subseteq A$ is a closed subset of $A$, then $B$ is also compact;
• (iii) $A$ is closed.

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Theorem: Interval nesting

Let $\{I_j{\}}_{j\in J}\subseteq \mathbb{R}$ be a collection of closed intervals such that $I_{j+1}\subseteq I_j$, and all $I_j$ are nonempty. Then their arbitrary intersection is nonempty, i.e., ${\bigcap }_{j=1}^{\infty }{I}_j\ne \varnothing$.

#wip Why?

Theorem: Compactness of intervals in \mathbb R

Any closed interval $[a,b]\subseteq \mathbb{R}$ is compact, meaning there exists a finite subcover of $[a,b]$.

Theorem: Boxes in R^n are compact

Theorem: Cantor intersection

Let $A_i\subseteq {\mathbb{R}}^n$ be a family of compact sets such that the intersection of any finite collection of $A_i$ is nonempty. Then their arbitrary intersection is nonempty, i.e., ${\bigcap }_{i=1}^{\infty }{A}_i\ne \varnothing$.

Connectedness §

Definition: Connected sets in metric spaces

Given a metric space $(X,d)$, a subset $A\subseteq X$ is said to be connected if for all other (nonempty) subsets $U,V\in X$ where

$$\overline{U}\cap V=U\cap \overline{V}=\varnothing ,$$

we have $U\cup V\ne A$. The sets $U,V$ which satisfy the equality above are said to be separated; a subset is connected if it is not the union of two (nonempty) separated sets.

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Bounded sets and functions §

Definition: Diameter of a bounded set in a metric space

Let $(X,d)$ be a metric space and $A\subseteq X$ be bounded. Then the diameter of $A$ is defined by

$$\text{diam}(A)=\text{sup}\{d(x,y)|x,y\in A\}.$$

Note that $A$ is bounded if and only if the diameter is finite.

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Proposition

Let $A\subseteq X$ be bounded. Then $\text{diam}(A)=\text{diam}(\overline{A})$, where $\overline{A}$ is the closure of $A$.

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Sequences and series §

Sequences §

Given a metric space $(X,d)$, a sequence, often denoted $(x_n{)}_{n\in \mathbb{N}}$, is a discrete function $\mathbb{N}\to \mathbb{X}$ which maps each natural number $n$ to some $x_n\in X$.

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Definition: Bounded sequences in metric spaces

Let $(X,d)$ be a metric space. We say a sequence $(x_n{)}_{n\in \mathbb{N}}$ in $X$ is bounded if there exist $x\in X$ and $r>0$ such that for all $n\in \mathbb{N}$, we have $d(x_n,x)<r$.

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Definition: Convergent sequence in a metric space

Given a metric space $(X,d)$, a sequence $(x_n{)}_{n\in \mathbb{N}}$ is called convergent in $X$ if there exists $x\in X$ such that

$${\forall }_{ϵ>0}{\exists }_{N\in \mathbb{N}}{\forall }_{n\in \mathbb{N}}[n\ge N⟹d(x_n,x)<ϵ].$$

In this case, we say $x$ is the limit of $(x_n{)}_{n\in \mathbb{N}}$ and write ${\lim }_{n\to \infty }{x}_n=x$. A sequence is divergent if it does not converge to some $x$.

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Definition: Sequence escaping to infinity in \mathbb R

We say a sequence $(x_n{)}_{n\in \mathbb{N}}$ in $\mathbb{R}$ escapes to infinity if for every $C\in \mathbb{N}$, there exists $N\in \mathbb{N}$ such that $x_n\ge C$ for all $n\ge N$.

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Definition: Limit superior of a sequence in \mathbb R

Let $(x_n{)}_{n\in \mathbb{N}}$ be a sequence $\mathbb{R}$, and let $A$ be the set of all sub-sequential limits of $(x_n{)}_{n\in \mathbb{N}}$. Then the limit superior, denoted $\lim \sup x_n$, is the value given by:

• $\sup A$, if $A$ is bounded and not empty;
• $\infty$, if there exists a sub-sequence of $(x_n{)}_{n\in \mathbb{N}}$ that escapes to infinity;
• $-\infty$, if the entire sequence $(x_n{)}_{n\in \mathbb{N}}$ escapes to $-\infty$.

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Proposition: Facts about convergent sequences in metric spaces

• (i) Convergent sequences converge to a unique limit.
• (ii) If a sequence converges, then it is also bounded.
• (iii) Every convergent sequence is Cauchy.

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Proposition: The set of sub-sequential limits is closed.

Let $A$ be the set of limits of subsequences of a sequence $(x_n{)}_{n\in \mathbb{N}}$ in $\mathbb{R}$. Then $A$ is closed.

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Proposition: Sequence characterization of limit points

Given a metric space $(X,d)$ and a subset $A\subseteq X$, if $x\in X$ is a limit point of $A$, then there exists a sequence $(x_n{)}_{n\in \mathbb{N}}$ in $A$ which converges to $x$.

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Proposition: Sequence characterization of closed sets

Let $(X,d)$ be a metric space and $A\subseteq X$. Then $A$ is closed if and only if every convergent sequence in $A$ converges to some $a\in A$.

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Cauchy sequences and complete metric spaces §

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Limit point and sequential compactness §

Definition: Limit point compact

A metric space $(X,d)$ is limit point compact if every infinite subset of $X$ has a limit point.

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Definition: Sequentially compact metric space

A metric space $(X,d)$ is called sequentially compact if every sequence has a convergent subsequence.

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Series §

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Continuous functions §

Definition: Continuous functions in metric spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. We say a function $f:X\to Y$ is:

• Continuous at $x_0\in X$ if for every $ϵ>0$, there exists $\delta >0$ such that

$$d_X(x,x_0)<\delta ⟹d_Y(f(x),f(x_0))<ϵ,$$

and $f$ continuous if this holds for all $x_0\in X$.

• Uniformly continuous if for all $ϵ>0$, there exists $\delta >0$ such that for any choice of $x,x_0\in X$, we have

$$d_X(x,x_0)<\delta ⟹d_Y(f(x),f(x_0))<ϵ.$$

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Definition: Homeomorphism, homeomorphic

A function between topological spaces $f:X\to Y$ is a homeomorphism if $f$ is continuous, bijective, and its inverse $f^{-1}:X\to Y$ is continuous.

$X,Y$ are homeomorphic if there exists a homeomorphism between them, and we write $X\cong Y$. Equivalently, we have $X\cong Y$ if there exists continuous functions $f:X\to Y$, $g:Y\to X$ such that $g\circ f={\text{id}}_X$ and $f\circ g={\text{id}}_Y$.

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Theorem: Extreme value theorem on a metric space

Let $(X,d_X)$ be a metric space and $K\subseteq X$ be a compact subset, and suppose $f:X\to \mathbb{R}$ is a continuous function. Then $f$ attains its maximum and minimum within $K$; that is, there exists $x_0\in K$ such that

$$f(x_0)=\underset{x\in X}{\sup }\{f(x)\}.$$

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Theorem: Continuity is equals uniform continuity on compact spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. If $f:X\to Y$ is continuous and $X$ is compact, then $f$ is uniformly continuous.

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Connectedness §

Differentiable functions §

Derivatives of real functions §

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Theorem:

Let $f:[a,b]\to \mathbb{R}$ be differentiable.

• If $f’(x)\ge 0$ for all $x\in [a,b]$, then $f$ is monotonically increasing.
• If $f’(x)=0$ for all $x\in [a,b]$, then $f$ is constant.

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Local extrema of real functions §

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Higher-order derivatives of real functions §

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Definition: Taylor polynomial

If $f:[a,b]\to \mathbb{R}$ is $n$-times differentiable at $x_0\in [a,b]$, then its $n$-th order Taylor polynomial centered at $x_0$ is the series defined as

$$T_{n,x_0}(x):=\sum _{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k.$$

Notice that for all $0\le k\le n$, we have $T_{n,x_0}(x_0)=f^{(k)}(x_0)$. Further, denote the approximation error or remainder as $x\to x_0$ by

$$R_n(x)=f(x)-T_{n,x_0}(x)=o(|x-x_0{|}^n),$$

where the final expression uses little-o notation.

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Theorem: Taylor approximation

Let $f:[a,b]\to \mathbb{R}$ be $(n=k+1)$-times continuously differentiable at some point $x_0\in [a,b]$, where $k>0$, and write $T_{n,x_0}$ for the $n$-th order Taylor polynomial of $f$ and $R_n(x)$ for the approximation error as $x\to x_0$. Then:

• (i) Peano form of the remainder. There exists a function $h_n:[a,b]\to \mathbb{R}$ defined by $h_n(x):=f(x)-(T_{n,x_0})(x)$ which satisfies

$$\underset{x\to x_0}{\lim }\frac{h_n(x)}{(x-x_0{)}^n}=0.$$

Thus, Taylor’s theorem says that the error term $o(|x-x_0{|}^n)$ decays as fast as some multiple of $|x-x_0{|}^{n+1}$, and in particular faster than any multiple of $|x-x_0{|}^n$.

• (ii) Cauchy form of the remainder. There exists some ${\xi }_C\in (x_0,x)$ such that

$$R_{k,C}(x)=\frac{f^{(k+1)}({\xi }_C)}{k!}(x-x_0)(x-{\xi }_C{)}^k.$$

• (iii) Lagrange form of the remainder. There exists some ${\xi }_L\in (x_0,x)$ such that

$$R_{k,L}(x)=\frac{f^{(k+1)}({\xi }_L)}{(k+1)!}(x-x_0{)}^{k+1}.$$

• (iv) Integral form of the remainder.

$$R_k(x)={\int }_{x_0}^x\frac{f^{(k+1)}(t)}{k!}(x-t{)}^{k}dt.$$

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Integration §

Development of the Darboux integral §

Definition: Partition

Given a bounded set $[a,b]\subset \mathbb{R}$, a partition $P=\{x_0,\dots ,x_n\}$ of $[a,b]$ is the collection of intervals made from a finite list of “sampling points” with $a=x_0<\cdots <x_n=b$.

A refinement of this partition is made by adding sampling points to generate new intervals. Two partitions of $P,P^{\prime }$ need not be comparable, but one can use their common refinement

$$P\cup P^{\prime }=\{x_0,\dots ,x_n,x_0^{\prime },\dots ,x_m^{\prime }\}.$$

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Definition: Darboux sums and integrals, Darboux integrable in a single variable

Let $f:[a,b]\to \mathbb{R}$ be a function, and let $P=\{x_0,\dots ,x_n\}$ be a partition of $[a,b]$. Then the upper and lower Darboux sums are defined by

$$U(f,P)=\sum _{k=1}^n(x_k-x_{k-1})\underset{t\in [x_{k-1},x_k]}{\sup }f(t)$$ $$L(f,P)=\sum _{k=1}^n(x_k-x_{k-1})\underset{t\in [x_{k-1},x_k]}{\inf }f(t),$$

respectively. Further, the upper and lower Darboux integrals are given by taking the infimum and supremum of all partitions of $[a,b]$, respectively:

$$U_f=\underline{{\int }_a^b}f=\inf \{U(f,P)|P\text{ is a partition of }[a,b]\}$$ $$L_f=\overline{{\int }_a^b}f=\sup \{L(f,P)|P\text{ is a partition of }[a,b]\}.$$

If $U_f=L_f$, then $f$ is said to be Darboux integrable.

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Theorem (Rudin 6.5): The upper Darboux sum is greater than or equal to the lower Darboux sum

Let $f:[a,b]\to \mathbb{R}$ be bounded. Then

$$L_f=\underset{\text{partitions }P}{\sup }L(f,P)\le U_f=\underset{\text{partitions }P}{\inf }U(f,P).$$

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Darboux and Riemann integration §

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