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 $ε$-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 \(1\).