Real Analysis

A Cartesian product of non-empty sets is non-empty.

\(f: A\rightarrow B\) \[f \subseteq A \times B \iff \forall x \in A,\; \exists !y \in B | (x,y) \in f\]

\(f: A\hookrightarrow B\) \[\forall x_1, x_2 \in A,\; f(x_1) = f(x_2) \implies x_1 = x_2 \]

\(f: A\twoheadrightarrow B\) \[\forall y \in B, \exists x \in A \mid f(x) = y\]

(injective and surjective) \[\forall y \in B,\; \exists !x \in A \mid f(x) = y\]

We say that two sets $A$ and $B$ have the same cardinality if there is a bijection $f: A\rightarrow B$; we then write $A\sim B$, which is the same as $|A| = |B|$.

Also, if there is an injective function $f:A\rightarrow B$, we say $|A|\leq |B|$. Or, equivalently, a surjection from $f: B\rightarrow A$.

A set $S$ is finite $|S| = {1,\ldots,n}$, for some $n\in\mathbb{N}$. Otherwise $S$ is infinite.

A set $S$ is Dedekind-infinite if there is a bijection from $S$ to a proper subset of itself. Otherwise $S$ is Dedekind-finite.

We say that a set $S$ is countable if $|S| \leq |\mathbb{N}|$. Otherwise $S$ is uncountable. If $S$ is countable and infinite we say that $S$ is countably infinite.

$\displaystyle\lim_{x \rightarrow a} f(x) = b$ means that "for any number $\varepsilon > 0$, there is a number $\delta(\varepsilon)$ such that $|f(x)-b| < \varepsilon$ whenever $|x-a|<\delta$"

A metric space is a pair $(X,d)$, where $X$ is a (non-empty) set and $d: X \times X \rightarrow [0,\infty)$ is a function, such that the following conditions hold for all $x,y,z \in X$:

1. $d(x,y) = 0 \iff x=y$
2. $d(x,y) = d(y,x)$
3. $d(x,y) + d(y,z) \geq d(x,z)\quad$ (triangle inequality)

A sequence in a set $X$ is a function from $\mathbb{Z}^+$ to $X$. \[\set{x_n}^\infty_{n=0}\]

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous if for every $x\in \mathbb{R}$ and every $\epsilon > 0$, there is a $\delta(x, \epsilon) > 0$ such that $|f(y) - f(x)| < \epsilon$ whenever $|y-x|<\delta(x,\epsilon)$.

For a point $x$ in a metric space $(X,d)$ and a number $\epsilon > 0$, define the $\epsilon$-ball \[B(x,\epsilon) = \set{y\in X: d(y,x) < \epsilon}\]

Let $(X,d)$ be a metric space, and consider $Y\subseteq X$. Define the interior \[\mathrm{Int}(Y) = \set{y \in Y: \exists \epsilon > 0\text{ such that }B(y,\epsilon)\subseteq Y} \]

Define the boundary: \[\mathrm{Bd}(Y) = X \backslash (\mathrm{Int}(Y)\cup \mathrm{Int}(Y^c))\]

A subset $Y$ in $(X,d)$ is open if $Y=\mathrm{Int}(Y)$

A subset $Y$ in $(X,d)$ is closed if $Y^c$ is open.

The closure of $Y$ is $\mathrm{Cl}(Y) = \mathrm{Int}(Y) \sqcup \mathrm{Bd}(Y)$.

$Y$ is dense if $\mathrm{Cl}(Y) = X$

Let $(X,d)$ be a metric space. An open neighbourhood of a point $x\in X$ is an open set $V\subseteq X$ such that $x\in V$. A neighbourhood of $x$ is a set $U\subseteq X$ such that there is an open neighbourhood $V$ of $x$ with $V\subseteq U$.

The set of open sets in a metric space $X$ is called the topology of $X$. \[\tau = \mathcal{O}(X)\]

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A function $f:X\rightarrow Y$ is continuous if for every $V\in \mathcal{O}(Y)$ we have $f^{-1}(V)\in\mathcal{O}(X)$.

A subset $Y\subseteq X$ satisfying the equivalent conditions above is bounded. (If $Y=X$, we say the metric space itself is bounded.)

A sequence $\{x_n\}_{n=0}^\infty$ in a metric space $(X,d)$ is Cauchy if \[ \forall\varepsilon>0\;\exists K(\varepsilon)\in\mathbb N\;:\; d(x_m,x_n)<\varepsilon\quad\text{whenever }m,n>K(\varepsilon). \]

A metric space $(X,d)$ is complete if every Cauchy sequence in $X$ converges to a point of $X$.

Two Cauchy sequences $\{a_n\}$ and $\{b_n\}$ in $(X,d)$ are equivalent if \[ \lim_{n\to\infty} d(a_n,b_n)=0. \]

Let $\overline X$ be the set of equivalence classes of Cauchy sequences in $X$. For classes $[a_n]$ and $[b_n]$ define \[ \overline d\bigl([a_n],[b_n]\bigr)\;=\;\lim_{n\to\infty} d(a_n,b_n). \] Then $(\overline X,\overline d)$ is called the completion of $X$.

For a vector space $V$ (over $\mathbb R$ or $\mathbb C$), a norm is a map \[ \|\cdot\|:V\to[0,\infty) \] such that for all $x,y\in V$ and $\lambda\in\mathbb R\text{ or }\mathbb C$:

1. $\|x\|=0\iff x=0$;
2. $\|\lambda x\|=|\lambda|\,\|x\|$;
3. $\|x+y\|\le\|x\|+\|y\|$.

A Banach space is a normed vector space that is complete in the metric induced by its norm.

An inner product space is a vector space $V$ together with $\langle\cdot,\cdot\rangle:V\times V\to\mathbb R\text{ or }\mathbb C$ satisfying

1. $\langle x,x\rangle>0$ for $x\ne0$;
2. $\langle x,y\rangle=\langle y,x\rangle$;
3. $\langle x+\lambda y,z\rangle=\langle x,z\rangle+\lambda\langle y,z\rangle$.

A Hilbert space is a complete inner product space.

A map $f:(X,d)\to(X,d)$ is a contraction if $\exists\,c\in(0,1)$ such that $d \left (f(x),f(y)\right )\le c\,d(x,y)$ for all $x,y\in X$.

For $X\subseteq\mathbb R$, a function $f:X\to\mathbb R$ is Lipschitz continuous if $\exists K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in X$. Such a constant $K$ is called a Lipschitz constant for $f$.

For $X\subseteq\mathbb R^{\,2}$, a function $f:X\to\mathbb R$ is Lipschitz continuous in the second variable if $\exists K>0$ such that \[|f(x,y_1)-f(x,y_2)|\le K\,|y_1-y_2|\quad\forall (x,y_1),(x,y_2)\in X.\]

A sequence of numbers $\{x_n\}_{n=1}^{\infty}\subset\mathbb{R}$ converges to $x$ if for every $\varepsilon>0$ there exists $K(\varepsilon)\in\mathbb{N}$ such that \[|x_n-x|<\varepsilon\quad\text{whenever }n\ge K(\varepsilon).\]

A sequence of functions $f_n:X\to\mathbb{R}$ converges pointwise to $f$ if for every $x\in X$ and every $\varepsilon>0$ there exists $K(x,\varepsilon)\in\mathbb{N}$ such that \[|f_n(x)-f(x)|<\varepsilon\quad\text{whenever }n\ge K(x,\varepsilon).\]

A sequence of functions $f_n:X\to\mathbb{R}$ converges uniformly to $f$ if for every $\varepsilon>0$ there exists $K(\varepsilon)\in\mathbb{N}$ such that \[|f_n(x)-f(x)|<\varepsilon\quad\text{for all }x\in X\text{ whenever }n\ge K(\varepsilon).\]

Let $B(X,\mathbb{R})$ be the set of bounded real‑valued functions on $X$. The uniform norm is defined by \[\|f\|_\infty=\sup_{x\in X}|f(x)|.\]

Let $f_n:[a,b]\to\mathbb{R}$ be Riemann‑integrable and $p\ge1$. We say $f_n\to f$ in $L^p$ if \[ \lim_{n\to\infty}\int_a^b |f_n(x)-f(x)|^p\,dx=0. \]

For a power series $\sum_{n=0}^{\infty}a_n x^n$ let \[ b=\limsup_{n\to\infty}|a_n|^{1/n},\qquad R=\frac1b\,(R\in[0,\infty]). \] The number $R$ is the radius of convergence. (If $b=0$ we set $R=\infty$, while if $b=\infty$ we set $R=0$.)

A topological space is a pair \(\left( X ,\tau \right)\), where \(X\) is a set and \(\tau\subseteq\mathcal{P}\!\left( X \right)\) satisfies

1. \(\varnothing , X \in \tau\);
2. if \(\{V_i\}_{i\in I}\subseteq\tau\) then \(\bigcup_{i\in I} V_i \in \tau\);
3. if \(V_1,V_2\in\tau\) then \(V_1\cap V_2\in\tau\).

The sets in \(\tau\) are open; their complements are closed.

For a topological space \(\left( X ,\tau \right)\) and \(Y\subseteq X\), the subspace topology on \(Y\) is \[ \tau\!\mid_Y=\{V\cap Y : V\in\tau\}. \]

In \(\left( X ,\tau \right)\) a subset \(C\subseteq X\) is closed if \(X\setminus C\in\tau\).

Let \(\left( X ,\tau \right)\) be a topological space.

• An open neighbourhood of \(x\in X\) is a set \(V\in\tau\) with \(x\in V\). A neighbourhood of \(x\) is any set containing an open neighbourhood of \(x\).
• For \(Y\subseteq X\) the interior is

\[ \operatorname{Int}\left( Y \right)=\{y\in Y : \exists V\in\tau,\; y\in V\subseteq Y\}. \]

For \(Y\subseteq X\) set \[ \operatorname{Bd}\left( Y \right)=X\setminus\!\left( \operatorname{Int}\left( Y \right)\cup\operatorname{Int}\left( X\setminus Y \right) \right),\qquad \operatorname{Cl}\left( Y \right)=\operatorname{Int}\left( Y \right)\cup\operatorname{Bd}\left( Y \right). \]

A sequence \(\{x_n\}_{n=1}^{\infty}\subseteq X\) converges to \(x\in X\) if for every \(V\in\tau\) with \(x\in V\) there exists \(K(V)\in\mathbb{N}\) such that \(x_n\in V\) whenever \(n\ge K(V)\).

For topological spaces \(\left( X ,\tau_X \right)\) and \(\left( Y ,\tau_Y \right)\), a function \(f:X\to Y\) is continuous if \(f^{-1}\!\left( V \right)\in\tau_X\) for every \(V\in\tau_Y\).

A topological space is Hausdorff if for every distinct \(x,y\in X\) there exist disjoint neighbourhoods \(V\in\operatorname{Nbhd}\!\left( x \right)\) and \(U\in\operatorname{Nbhd}\!\left( y \right)\).

Let \(\left( X ,\tau \right)\) be a topological space.

• A base \( \mathcal{B}\subseteq\tau \) satisfies: every \(V\in\tau\) can be written \(V=\bigcup_{i\in I} B_i\) with \(B_i\in\mathcal{B}\).
• A local base at \(x\in X\) is a collection \(\mathcal{L}_x\subseteq\tau\) of neighbourhoods of \(x\) such that for every neighbourhood \(U\) of \(x\) there is \(V\in\mathcal{L}_x\) with \(V\subseteq U\).

Given \(S\subseteq\mathcal{P}\!\left( X \right)\), let \(\mathcal{B}\) be the set of all finite intersections of elements of \(S\) (including \(X\)). The topology generated by \(S\) is \(\tau\!\left( S \right)=\{V\subseteq X : V\text{ is a union of sets in }\mathcal{B}\}\). The collection \(S\) is a subbase for \(\tau\!\left( S \right)\).

A space is first countable if every point has a countable local base. It is second countable if it possesses a countable base.

A topological space is separable if it contains a countable dense subset.

A directed set is a set \(\Lambda\) with a relation \(\le\) such that

1. \(i\le i\) for all \(i\in\Lambda\);
2. \(i\le j\le k\Rightarrow i\le k\);
3. for \(i,j\in\Lambda\) there exists \(m\in\Lambda\) with \(i\le m\) and \(j\le m\).

A net in \(X\) is a function \(\Lambda\to X\) where \(\Lambda\) is directed; we write \(\{x_\lambda\}_{\lambda\in\Lambda}\).

A net \(\{x_\lambda\}\) converges to \(x\in X\) if for every neighbourhood \(V\) of \(x\) there exists \(\alpha\in\Lambda\) such that \(x_\lambda\in V\) whenever \(\lambda\ge\alpha\).

For topologies \(\tau\subseteq\sigma\) on the same set \(X\), \(\tau\) is coarser and \(\sigma\) is finer.

A bijection \(f:X\to Y\) between topological spaces is a homeomorphism if both \(f\) and \(f^{-1}\) are continuous. Spaces that admit a homeomorphism are homeomorphic.

A space \(\left( X ,\tau \right)\) is connected if it cannot be written as the union of two disjoint non‑empty open sets. A subset \(Y\subseteq X\) is connected in the subspace topology.

A space is path‑connected if for all \(x,y\in X\) there exists a continuous map \(f:[0,1]\to X\) with \(f(0)=x\) and \(f(1)=y\).

For topological spaces \(\left( X ,\tau_X \right)\) and \(\left( Y ,\tau_Y \right)\) the product topology on \(X\times Y\) is generated by the base \(\{U\times V : U\in\tau_X,\; V\in\tau_Y\}\).

Given \(\{(X_i,\tau_i)\}_{i\in I}\), the box topology on \(\prod_{i\in I} X_i\) has base \[ \left\{ \prod_{i\in I} U_i : U_i\in\tau_i\text{ for every }i\in I \right\}. \]

A topological space \(\left( X ,\tau \right)\) is compact if for every open cover \[ X \;=\;\bigcup_{i\in I} V_i ,\qquad V_i\in\tau , \] there exists a finite sub‑cover \(V_{i_1},\ldots,V_{i_n}\) with \[ X \;=\;\bigcup_{k=1}^{n} V_{i_k}. \] A subset \(Y\subseteq X\) is compact when it is compact in the subspace topology.

A space \(\left( X ,\tau \right)\) is sequentially compact if every sequence in \(X\) has a convergent subsequence. The notion for subsets uses the subspace topology.

For metric spaces \(\left( X ,d_X \right)\) and \(\left( Y ,d_Y \right)\) a function \(f:X\to Y\) is uniformly continuous if \[ \forall\varepsilon>0\;\exists\delta(\varepsilon)>0\text{ such that }d_Y\!\left( f(x),f(x') \right)<\varepsilon \text{ whenever }d_X\!\left( x,x' \right)<\delta(\varepsilon). \]

A metric space \(\left( X ,d \right)\) is totally bounded if for every \(\varepsilon>0\) there exist points \(x_1,\ldots,x_n\in X\) such that \[ X \;=\;\bigcup_{k=1}^{n} B\!\left( x_k,\varepsilon \right). \]

Let \(\left( X ,d_X \right)\), \(\left( Y ,d_Y \right)\) be metric spaces and \(S\subset C\!\left( X ,Y \right)\).

• Pointwise equicontinuous: \(\forall x\in X,\;\forall\varepsilon>0,\;\exists\delta(x,\varepsilon)>0\) such that \(d_Y\!\left( f(x),f(x') \right)<\varepsilon\) for every \(f\in S\) whenever \(d_X\!\left( x,x' \right)<\delta(x,\varepsilon)\).
• Uniformly equicontinuous: \(\forall\varepsilon>0,\;\exists\delta(\varepsilon)>0\) satisfying the same inequality for all \(x,x'\in X\) and all \(f\in S\).

A set \(A\subset F\!\left( X ,k \right)\) (functions from \(X\) to a field \(k\)) is an algebra if it is a vector space under pointwise operations and closed under pointwise multiplication. It is unital when it contains the constant function \(1\).