Title: Theory / Important / Math / Analysis / Zorich{ / 01 Limits and continuity; / 2 Several variables;
Tags: fromImplication examplesSwitchExample1 closed fromMath mathToSee omitted récupéré

Name: A point
Definition: An element of \(\mathbb R^m\)
Page: 410
Chapter: 7
Section: 1.1
Variables: \(m\in\mathbb N\)
Denoted by: \(x=(x^1,\dots,x^m)\)

Name: The \(i\)-th coordinate of \(x\)
Definition: \(x^i\)
Page: 410
Chapter: 7
Section: 1.1
Variables: \(m\in\mathbb N\),
\(x=(x^1,\dots,x^m)\) a point

Name: The euclidean distance on \(\mathbb R^n\)
Definition: [$]\defFun{(\mathbb R^n)^2}{\mathbb R}{((x_1^1,\dots,x_1^m),(x_2^1,\dots,x_2^m))}{\sqrt{\sum (x_1^i-x_2^i)^2}}[/$].
Page: 410
Chapter: 7
Section: 1.1
Variables: \(m\in\mathbb N\),
Context: \(\mathbb R^n\)
Notation: \(d\)
DefType: The distance

Name: A positive function
Definition: A function \(S\times S\to \mathbb R^{\ge 0}\)
Page: 410
Chapter: 7
Section: 1.1
Variables: \(S\) a set,

Name: A definite function
Definition: A function \(d:S\times S\to \mathbb R^{\ge 0}\) such that \(d(x,y)=0\) is equivalent to \(x=y\).
Page: 410
Chapter: 7
Section: 1.1
Variables: \(S\) a set,

Name: A symmetric function
Definition: A function \(d:S\times S\to X\) such that \(d(x,y)=d(y,x)\).
Page: 410
Chapter: 7
Section: 1.1
Variables: \(S\) a set,

Name: A function satisfying the triangular inequality
Definition: A function \(d:S\times S\to \mathbb R^{\ge 0}\) such that \(d(x,y)\le d(x,z)+d(y,z)\).
Page: 410
Chapter: 7
Section: 1.1
Variables: \(S\) a set,

Name: A metric
Definition: A positive definite symmetric function satisfying the triangular inequality
Page: 410
Chapter: 7
Section: 1.1
Variables: \(S\) a set,
Name2: A distance

Name: A metric space
Definition: A set together with a fixed metric.
Page: 410
Chapter: 7
Section: 1.1

Name: The ball with center \(a\) of radius \(\delta\)
Definition: The set of \(x\in\mathbb R^m\) such that \(d(a,x)<\delta\).
Page: 411
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(a\in\mathbb R^m\),
\(\delta>0\)
Notation: \(B(a,\delta)\)
Name2: The \(\delta\)-neighborhood of the point \(a\)
Extra: Open ball

Name: An open set of \(\mathbb R^m\)
Definition: A set \(G\subseteq \mathbb R^m\) such that for each \(x\in G\), there exists \(\delta_x\) such that \(B(x;\delta_x)\subseteq G\).
Page: 411
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
Index: 2, 1, 2, 3, 4
Kind: Def, Exs
Example: \(\mathbb R^m\), \(\emptyset\)
Example2: \(B(a;\delta)\) for arbitrary \(a,\delta\), \(\{x\mid d(a,x)>\delta\}\)

Name: A closed set of \(\mathbb R^m\)
Definition: which is the complement of an open set
Page: 411, 6
Chapter: 7, 0
Section: 1.2, 1,
Variables: \(m\in\mathbb N\),
Index: 3
Kind: Def, Stillwell
Definition2: such that all infinite sequences in \(S\) have a converging subsequence in \(R\)
Conjdef: A set \(S\)
Extra: It has many other definition. Keeping the one of the book

Name: The closed ball with center \(a\) of radius \(\delta\)
Definition: The set of \(x\in\mathbb R^m\) such that \(d(a,x)\le\delta\).
Page: 411
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(a\in\mathbb R^m\),
\(\delta>0\)
Notation: \(\overline B(a,\delta)\)
Name2: The \(\delta\)-neighborhood of the point \(a\)

Name: The class of open sets
Page: 411, 58
Section: 1.2, 1.3
Chapter: 7 Zorich, 3 Ahlfors
Index: 1, 5
Kind: Prop, theo
Under: unions
Under2: intersections
Condition2: finite but not arbitrary
Under3: taking a component

Name: A neighborhood of \(x\)
Definition: An open set containing \(x\)
Page: 411
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(x\in\mathbb R^m\)
Index: 4
Kind: Def
Example: A \(\delta\)-neighborhood

Name: An interior point of \(E\)
Definition: A point which admits a neighborhood contained in \(E\)
Page: 411
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Index: 5
Kind: Def

Name: An exterior point of \(E\)
Definition: A point which admits a neighborhood disjoint from \(E\)
Page: 411
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Index: 5
Kind: Def

Name: A boundary point of \(E\)
Definition: A point which is neither interior nor exterior
Page: 411
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Index: 5
Kind: Def

Name: A limit point of \(E\)
Definition: For any neighborhood \(O(a)\) of \(a\), the intersection \(E\cap O(a)\) is infinite
Page: 413
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Index: 6
Kind: Def

Name: The closure of \(E\)
Definition: The union of \(E\) and its limit point
Page: 413
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Notation: \(\overline E\)
Index: 6
Kind: Def

Name: The sphere of center \(a\) and radius \(r\)
Definition: The set of points \(x\) whose distance to \(a\) is \(r\)
Page: 412
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(a\in\mathbb R^m\),
\(r>0\)
Notation: \(S(a;r)\)
Kind: Ex

Smaller0: \(F\) closed in \(\mathbb R^m\)
Greater0: \(\overline F=F\)
Page: 413
Chapter: 7
Section: 1.2
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Index: 2
Kind: Prop
Smaller to greater: iff

Name: A compact set
Definition: A set \(E\subseteq\mathbb R^m\) such that, from every covering of \(K\) by sets that are open in \(\mathbb R^m\), one can exctract a finite covering
Page: 413
Chapter: 7
Section: 1.3
Variables: \(m\in\mathbb N\),
Index: 2
Kind: Prop
Example: \(m\)-dimensional intervals

Name: An \(m\)-dimensional interval
Definition: A set of \(x\in\mathbb R^m\) with \(x^i\in[a^i,b^i]\), for some \(a,b\in\mathbb R^m\) fixed
Page: 413
Chapter: 7
Section: 1.3
Variables: \(m\in\mathbb N\),
Index: 13
Kind: Ex
Name2: An \(m\)-dimensional block
Name3: An \(m\)-dimensional parallelepiped

Name: The diameter of \(E\)
Definition: \(\text{sup}_{x_1,x_2\in E}d(x_1,x_2)\)
Page: 414
Chapter: 7
Section: 1.3
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Notation: \(d(E)\)
Index: 9
Kind: Def

Name: A bounded space
Definition: A space whose diameter is finite
Page: 414
Chapter: 7
Section: 1.3
Variables: \(m\in\mathbb N\),
Index: 10
Kind: Def

Smaller0: \(E\) is compact
Greater0: bounded
Page: 415
Chapter: 7
Section: 1.3
Variables: \(m\in\mathbb N\),
\(E\subseteq\mathbb R^m\)
Index: 5
Kind: Prop
Smaller to greater: iff
Prefix Greater: \(E\) is
Greater1: closed

Name: A bounded mapping
Definition: A function \(f:X\to\mathbb R^n\) such that \(f(X)\) is bounded in \(\mathbb R^n\)
Page: 416
Chapter: 7
Section: 2.1
Variables: \(X\) a set,
\(n\in\mathbb R\)
Index: 2
Kind: Def

Name: An ultimately bounded mapping
Definition: A function \(f:X\to\mathbb R^n\) such that \(f(B)\) is bounded in \(\mathbb R^n\) for some \(B\in\mathcal B\)
Page: 416
Chapter: 7
Section: 2.1
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\)
\(n\in\mathbb R\)
Index: 3
Kind: Def

Definition: For any ball \(B(A;r)\subseteq \mathbb R^n\), there exists \(B\in\mathcal B\) such that \(f(B)\subseteq \mathbb R^n\setminus B(A;r)\)
Page: 419
Chapter: 7
Section: 2.1
Variables: \(X\) a set with basis \(\mathcal B\),
\(f:X\to\mathbb R^n\)
Notation: \(f(x)\to\infty\) in the base \(\mathcal B\)

Question: Example of 2-ary function which admits distinct limits depending on the path used.
Answer: \(f(x,y)=\frac{xy}{x^2+y^2}\) for \((x,y)\ne 0\), \(0\) otherwise.
Limit \((x,y)\to 0\) with \(x=0\) is 0.
Limit \((x,y)\to 0\) with \(x=y\) is 1.
Page: 419
Chapter: 7
Section: 2.1
Index: 2
Kind: Ex

Name: A pathwise connected subset of \(\mathbb R^m\)
Definition: A set \(E\subseteq\mathbb R^m\) such that, for each \(x_0,x_1\in E\) there exists a path \(\Gamma:I\to E\) with support in \(E\) and endpoints at these points.
Page: 423
Chapter: 7
Section: 2.2
Index: 9
Kind: Def

Smaller0: \(E\) connected
Greater0: \(f\)'s image contains \([f(a_0),f(a_1)]\)
Page: 424
Chapter: 7
Section: 2.2
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R\),
\(a_0,a_1\in E\),
Index: d)
Smaller1: \(f\) continuous

Smaller0: \(E\) compact
Greater0: \(f\) uniformly continuous
Page: 424
Chapter: 7
Section: 2.2
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R^n\)
Smaller1: \(f\) continuous
Greater1: \(f\) is bounded, (assumes its min and max if \(n=1\))

Question: Omitted
Answer: notions of linear algebra (at least the one in Linear Algebra done right).
In particular, I omit:
-basic definition of spaces,
-linearily,
-basis,
-the vector space of linear functions,
-norm,
-inner product
-orthogonality,
-cartesian coordinate
-relation between all of this and matrices/vectors
Page: 428
Chapter: 8
Section: 1.2

Definition: [$]\norm{f_1(x)}_{Y_1}=o(\norm{f_2(x)}_{Y_2})[/$] over the base \(\mathcal B\)
Page: 430
Chapter: 8
Section: 1.3
Variables: \(X,Y_1,Y_2\) normed vector spaces.
\(\mathcal B\) a normed vector space of \(X\)
\(f_i:X\to Y_i\)
Notation: \(f_1=o(f_2)\) over \(\mathcal B\)

Definition: [$]\norm{f_1(x)}_{Y_1}=O(\norm{f_2(x)}_{Y_2})[/$] \(\mathcal B\)
Page: 430
Chapter: 8
Section: 1.3
Variables: \(X,Y_1,Y_2\) normed vector spaces.
\(\mathcal B\) a normed vector space of \(X\)
\(f_i:X\to Y_i\)
Notation: \(f_1=O(f_2)\) over \(\mathcal B\)

Smaller0: \(f_1=O(f_2)\) over \(\mathcal B\)
Greater0: \(f_1^i=O(f_2)\) over the base \(\mathcal B\) for \(i=1,\dots,n_1\)
Page: 430
Chapter: 8
Section: 1.3
Variables: \(X,Y_1,Y_2\) normed vector spaces.
\(\mathcal B\) a normed vector space of \(X\)
\(f_i:X\to Y_i\)
\((e_i^1,\dots,e_i^{n_i})\) a basis of \(Y_i\)
Smaller to greater: iff

Smaller0: \(f_1=o(f_2)\) over \(\mathcal B\)
Greater0: \(f_1^i=o(f_2)\) over the base \(\mathcal B\) for \(i=1,\dots,n_1\)
Page: 430
Chapter: 8
Section: 1.3
Variables: \(X,Y_1,Y_2\) normed vector spaces.
\(\mathcal B\) a normed vector space of \(X\)
\(f_i:X\to Y_i\)
\((e_i^1,\dots,e_i^{n_i})\) a basis of \(Y_i\)
Smaller to greater: iff

Name: An Euclidean space
Definition: A vector space \(\mathbb R^m\) with inner product \(\langle x^i_1,x^i_2\rangle\mapsto {\sum (x^i_1x^i_2)}\)
Page: 431
Chapter: 8
Section: 1.3

Name: A limit of \(f\) over \(\mathcal B\)
Definition: \(A=(A^1,\dots,A^n)\) and \(f^i\) converges to \(A^i\) over \(\mathcal B\)
Page: 416
Chapter: 7
Section: 2.1
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(n\in\mathbb R\),
\(f:X\to\mathbb R^n\)
Notation: \(\lim_{\mathcal B}f(x)\)
Index: 1, 1', 1''
Kind: Def
Definition2: for all neighborhood \(V(A)\) of \(A\), there exists \(B\in \mathcal B\) such that \(F(B)\subseteq V(A)\)
Conjdef: equivalently
Definition3: for all \(\epsilon>0\), there exists \(B\in\mathcal B\) such that for all \(x\in B\), \(d\left(f(x),A\right)<\epsilon\)
Definition4: \(\lim_{\mathcal B}(d(f(x),A))=0\).
DefType: \(A\in\mathbb R^n\)

Name: An Euclidean space
Definition: finite-dimensional
Page: 431, 48
Chapter: 8, 10
Section: 1.4, 1.3
Definition2: real
Conjdef: A vector space
Definition3: with inner product