Title: Theory / Unimportant / Math / Analysis / Real Pugh/{ / 1 - Real numbers;
Tags: fromMath mathToDef examplesSwitchExample1 closed hasSeenConjDef toBinaryFunction

Name: Cauchy condition
Definition: \(\forall \epsilon>0\exists N\in\mathbb N\). such that if \(n,k\ge N\) then \(|a_n-a_k|<\epsilon\)
Page: 018
Chapter: 2
Variables: \((a_n)\in \mathbb R\) a sequence.

Name: A set with the Archimedian property
Definition: A set \(S\supseteq\mathbb N\) such that, for all \(s\in S\), there exists \(n\in\mathbb N\) such that \(n\ge s\).

Name: Ordered field without Archimedian property
Example: Rational functions with \(p<q\) if, for all \(x\) great enough, \(p(x)<q(x)\).

Name: The triangle inequality
Definition: \[|x+y|\le |x|+|y|\].
Page: 024
Chapter: 1
Section: 3
Variables: \(x,y\in\mathbb R^m\).

Name: A box
Definition: \(\Pi_{i=1}^n [a_i,b_i]\).
Page: 026
Chapter: 1
Section: 3

Name: The unit ball of dimension \(m\)
Definition: \(\{x\in\mathbb R^m: |x|\le 1\}\).
Page: 026
Chapter: 1
Section: 3
Variables: \(m\in\mathbb N\).
Notation: \(B^m\)

Name: The unit sphere in \(\mathbb R^m\)
Definition: \(\{x\in\mathbb R^m: |x|= 1\}\).
Page: 026
Chapter: 1
Section: 3
Variables: Let \(m\in\mathbb N\).
Notation: \(S^{m-1}\)

Name: Convex subset of \(\mathbb R^m\)
Definition: For all \(x,y\in E\), the line between \(x\) and \(y\) is a subset of \(E\).
Page: 026
Chapter: 1
Section: 3
Variables: \(E\subseteq \mathbb R^m\).

Definition: The set of \(sx+ty\) where \(s+t=1\).
Page: 027
Chapter: 1
Section: 3
Variables: \(x,y\in\mathbb R^m\).
Context: \(\mathbb R^m\)
Notation: \([x,y]\)

Definition: The set of continuous real valued functions defined on \([a,b]\).
Page: 028
Chapter: 1
Section: 3
Variables: \(a\le b\in\mathbb R\).
Notation: \(C([a,b],\mathbb R)\)

Page: 028
Section: 3
Notation: \(\langle f,g\rangle \)
Variables: \(a\le b\)
\(f, g\in C([a,b],\mathbb R)\).
Chapter: 1
Name: The inner product of \(f\) and \(g\)
Definition: [$]\int_a^b f(x)g(x)\dd x[/$].
Context: Analysis

Name: Denumerable sets
Page: 34, 035
Section: 4
Chapter: 1
Index: 13, 18
Kind: Theo, Corollary
Under: union
Condition: denumerable
Under2: subset
Under3: product
Condition3: finite

Smaller0: there exists \(f_i:A_i\to A_{1-i}\) injective
Greater0: the cardinalities of \(A\) and \(B\) are equals
Page: 036
Chapter: 1
Section: 5
Variables: \(A_0,A_1\) sets
Index: 21
Kind: Theorem
Hide middle: yes
Name: Schroeder-Bernstein theorem

Name: A Dedekind Cut
Definition: \(A\cup B=\mathbb Q\)
Page: 012
Chapter: 2
Kind: Definition
Definition2: \(x<y\) and \(y\in A\) implies \(x\in A\)
Conjdef: A pair \((A,B)\) with
Definition3: \(A\) without greatest element.
Denoted by: \(A|B\)

Name: A rational function
Definition: \(\frac{p(x)}{q(x)}\) with \(p\) and \(q\) polynomials.

Smaller0: for each \(\epsilon>0\), \(|x-y|\le \epsilon\)
Greater0: \(x=y\)
Page: 021
Chapter: 1
Section: 2
Index: 8
Kind: Theorem
Smaller to greater: iff
Name: The \(\epsilon\)-principle

Smaller0: \(\langle x,y \rangle\)
Greater0: \(||x|| ||y||\)
Page: 023
Chapter: 1
Section: 3
Variables: \(x,y\in\mathbb R^m\).
Index: 9
Kind: Theorem
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Name: The scalar product of \(x\) and \(y\)
Definition: \(x_1y_1+\dots+x_ny_n\).
Page: 022
Chapter: 1
Section: 3
Variables: \(x,y\in\mathbb R^n\).
Notation: \(x\cdot y\)
Name2: The dot product
Intuition: product whose result is a scalar, denoted using a dot
product of the length of the projection of one vector on the other line

Definition: \(\sqrt{\langle x,x\rangle}\).
Page: 023
Chapter: 1
Section: 3
Variables: \(x\in\mathbb R^n\).
Notation: \(|x|\)

Name: The integer lattice
Definition: the set \(\mathbb Z^m\) considered as a subset of \(\mathbb R^m\).
Page: 024
Chapter: 1
Section: 3
Variables: \(m\in\mathbb N\).

Smaller0: \(S\) is infinite
Greater0: a denumerable subset
Page: 033
Chapter: 1
Section: 4
Variables: \(S\) be a set.
Index: 12
Kind: Theorem
Smaller to greater: iff
Prefix Greater: \(S\) contains

Name: A continuous function
Definition: For each \(\epsilon>0\) and each \(x\in[a,b]\) there is a \(\delta>0\) such that \(t\in[a,b]\) and \(|t-x|<\delta\implies |f(t)-f(x)|<\epsilon\).
Page: 038
Chapter: 1
Section: 6

Question: The values of \(f\) on \([a,b]\) form
Answer: a bounded subset of \(\mathbb R\).
Page: 038
Chapter: 1
Section: 6
Variables: \(f:[a,b]\to\mathbb R\) continuous.
Index: 22
Kind: Theorem

Question: The bounds of \(f\) on \([a,b]\) belongs to
Answer: the range of \(f\) on \([a,b]\).
Page: 039
Chapter: 1
Section: 6
Variables: \(f:[a,b]\to\mathbb R\) continuous.
Index: 23
Kind: Theorem

Question: \(f\) defined on an interval \([a,b]\) takes on
Answer: all intermediate values.
Page: 040
Chapter: 1
Section: 6
Variables: \(f:[a,b]\to\mathbb R\) continuous.
Index: 24
Kind: Theorem

Name: A norm not coming from an inner product
Page: 028
Chapter: 1
Section: 3
Example: max.

Smaller0: \((a_i)\) is a Cauchy sequence
Greater0: \((a_i)\) converges to a limit in \(\mathbb R\)
Page: 019
Chapter: 1
Section: 2
Variables: \((a_i)\in\mathbb R\).
Smaller to greater: iff

Smaller0: \(f\) continuous
Greater0: bounded
Page: 41
Chapter: 1
Section: 6
Variables: \(f:I\to\mathbb R\)
Index: 25
Kind: Theo
Prefix Greater: \(f\)
Smaller1: \(f\) defined on a closed interval
Greater1: achieve every values between minimum and maximum, including them
Greater2: is uniformly continuous