Title: Theory / Important / Math / Analysis / Zorich{ / 10 More general point of view
Tags: hasSeenConjDef defToAlgebra fromMath interval toBinaryFunction

Page: 91
42
Chapter: 3,Done right
10 , Zorich
Section: E,
1
Set: a tuple associating to each \(i\in\mathcal I\) an element \(v_i\in V_i\)
Variables: \(\mathbb F\) a field,
\(\mathcal I\) an index set
\((V_i)_{i\in\mathcal I}\) be \(\mathbb F\) vector spaces
Notation: \(\prod_{i\in\mathcal I} V_i\)
Kind: ,
Ex,
Index: 71, 73, 76
+: point-wise addition
0: tuple of 0
Typ: The vector space
×: point-wise scalar multiplication
Field-base: \(\mathbb F\)
Dimension: \(\prod_{i\in\mathcal I}\text{dim I}\)

Definition: The set of sequences \((x^i)_{i\in\mathbb N}\) with \(x_i\) real or complex.
Page: 41
Chapter: 10
Section: 1
Context: Analysis
Notation: \(\ell\)
Index: 2
Kind: Ex

Definition: The subset of \(\ell\) with finite support
Page: 41
Chapter: 10
Section: 1
Notation: \(\substack{\ell\\0}\)
Index: 2
Kind: Ex

Name: The support of \(x\)
Definition: The set of \(i\in\mathbb N\) such that \(x^i\ne 0\).
Page: 41
Chapter: 10
Section: 1
Variables: \(x=(x^i)\in\ell\)
Kind: Ex

Definition: The set of real or complex functions whose domain is \([a,b]\)
Page: 42
Chapter: 10
Section: 1
Variables: \(a\le b\)
Context: Analysis
Notation: \(F[a,b]\)
Index: 3
Kind: Ex

Definition: [$]\defFun{[a,b]}{\{0,1\}}{x}{\substack{1\text{ if $x=\tau$}\\0 \text{ othewise}}}[/$]
Page: 42
Chapter: 10
Section: 1
Variables: \(a\le\tau\le b\)
Notation: \(e_\tau\)
Index: 3
Kind: Ex
Extra: checked

Name: A norm
Definition: nondegenerate (i.e. [$]\norm{x}=0[/$] iff \(x=0\))
Page: 42
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field
\(X\) a \(\mathbb F\)-vector space
Index: 1
Kind: Def
Definition2: homogeneous (i.e. [$]\norm{\lambda x}=\abs{\lambda}\norm{x}[/$])
Conjdef: A function \(X\to\mathbb R\) which is
Definition3: satisfies the triangle inequality (i.e. [$]\norm{x+y}\le\norm{x}+\norm{y}[/$])
Denoted by: [$]\norm{\ }[/$]

Name: A nondegenerate function
Definition: A function \(f:X\to\mathbb R\) such that \(f(x)=0\iff x=0\)
Page: 42
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field
\(X\) a \(\mathbb F\)-vector space
Index: 1
Kind: Def

Name: A homogeneous function
Definition: A function \(f:X\to\mathbb R\) such that [$]f(\lambda x)=\abs{\lambda}f(x)[/$]
Page: 42
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field
\(X\) a \(\mathbb F\)-vector space
Index: 1
Kind: Def

Name: A function satisfying the triangle inequality
Definition: A function \(f:X\to\mathbb R\) such that \(f(x+y)\le f(x)+f(y)\) for each \(x,y\in X\)
Page: 42
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field
\(X\) a \(\mathbb F\)-vector space
Index: 1
Kind: Def

Name: A normed vector space
Definition: A vector space and a norm on it.
Page: 42
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field
Index: 2
Kind: Def

Name: The norm of \(x\)
Page: 42
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field,
[$](X,\norm{\ })[/$] a normed \(\mathbb F\)-vector space,
\(x\in X\)
Notation: [$]\norm{x}[/$]
Index: 3
Kind: Def
Typ: Nonnegative

Smaller0: \(\parallel\sum x_i\parallel\)
Greater0: \(\sum\parallel x_i\parallel\)
Page: 43
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field,
\((X,\parallel\ \parallel)\) a normed \(\mathbb F\)-vector space,
\(x_1,\dots,x_n\in X\)
Index: 3
Kind: Def
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)
Hide sides: yes

Smaller0: [$]\abs{\norm{x_1}-\norm{ x_2}}[/$]
Greater0: [$]\norm{ x_1-x_2}[/$]
Page: 43
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field,
\((X,\parallel\ \parallel)\) a normed \(\mathbb F\)-vector space,
\(x_1,x_2\in X\)
Index: 3
Kind: Def
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Page: 43
Section: 1.2
Notation: \(d\)
Variables: \(\mathbb F\) a field,
\((X,\parallel\ \parallel)\) a normed \(\mathbb F\)-vector space,
Chapter: 10
Name: The metric of \(X\)
Definition: The function [$]\defFun{X^2}{\mathbb R^{\ge0}}{(x_1,x_2)}{\parallel x_1- x_2\parallel}[/$]

Question: \(d(x_1+x,x_2+x)\)
Answer: \(=d(x_1,x_2)\)
Page: 43
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field,
\((X,\parallel\ \parallel)\) a normed \(\mathbb F\)-vector space,
\(x,x_1,x_2\in X\)

Question: \(d(\lambda x_1,\lambda x_2)\)
Answer: [$]=\abs{\lambda} d(x_1,x_2)[/$]
Page: 43
Chapter: 10
Section: 1.2
Variables: \(\mathbb F\) a field,
\((X,\parallel\ \parallel)\) a normed \(\mathbb F\)-vector space,
\(x,x_1,x_2\in X\)

Name: A complete normed vector space
Definition: A normed vector space whose distance is complete.
Page: 43, 44
Chapter: 10
Section: 1.2
Index: 4, 5, 7, 8
Kind: Def, Exs
Example: \(\mathbb R^n_p\) for \(n,p\ge 1\)
Name2: A Banach space
Example2: \(\ell _p\) for \(p\ge 1\)
Example3: \(C[a,b]\) with \(\max\) norm

Name: The \(p\)-norm
Definition: [$]\defFun{X_i}{\mathbb F}{x}{\left(\sum_{i\in I}^n\norm{x^i}_{X_i}^p\right)^{\frac 1p}}[/$] if it converges. Undefined otherwise
Page: 43, 44
Chapter: 10
Section: 1.2
Variables: \(n\in\mathbb N\),
\(\mathbb F\) a field
\(I\) a countable indexing family
[$](X_i,\norm{ }_{X_i})_{i\in I}[/$] a \(\mathbb F\)-normed vector spaces
\(x^i\in X_i\)
\(x=(x^i)_{i\in I}\)
Notation: [$]\norm{\cdot}_p[/$]
Index: 5, 6, 7
Kind: Ex
Typ: A norm, assuming \(p\ge1\)
Assuming: \(p>0\)

Definition: \(\text{sup}\{\parallel x^i\parallel_{X_i} \mid 1\le i\le n\}\) if the \(x^i\)'s are bounded. Undefined otherwise
Page: 43, 44
Chapter: 10
Section: 1.2
Variables: \(n\in\mathbb N\),
\(\mathbb F\) a field
\(I\) a countable indexing family
\((X_i,\parallel \parallel_{X_i}))_{i\in I} \) \(\mathbb F\)-normed vector spaces
\(x^i\in X_i\)
\(x=(x^i)_{i\in I}\)
Notation: [$]\norm{x}_\infty[/$]
Index: 5, 7
Kind: Ex

Question: [$]\lim_{p\to\infty}\norm{x}_p[/$]
Answer: equals [$]\norm{x}_\infty[/$]
Page: 43
Chapter: 10
Section: 1.2
Variables: \(n\in\mathbb N\),
\(\mathbb F\) a field
\(I\) a countable indexing family
\((X_i,\parallel \parallel_{X_i})_{i\in I}\) \(\mathbb F\)-normed vector spaces
\(x^i\in X_i\)
\(x=(x^i)_{i\in I}\)
\(p\ge 1\)
Index: 5, 7
Kind: Ex

Smaller0: [$]\norm{x}_p[/$]
Greater0: [$]\norm{x}_1[/$]
Page: 43
Chapter: 10
Section: 1.2
Variables: \(n\in\mathbb N\),
\(\mathbb F\) a field
\(I\) a countable indexing family
[$](X_i,\norm{\ }_{X_i})_{i\in I} [/$] \(\mathbb F\)-normed vector spaces
\(x^i\in X_i\)
\(x=(x^i)_{i\in I}\)
\(p\ge 1\)
Index: 5, 7
Kind: Ex
Smaller to greater: \(\le\)
Assuming: those numbers are defined
Greater to smaller: \(\ge\)
Smallest: [$]\norm{x}_\infty[/$]
Greatest: [$]\abs{l}\norm{x}_\infty[/$]

Page: 44
Chapter: 10
Section: 1.2
Set: The set of elements of \(x=(x^i)_{i\in I}\) of finite norm
Variables: \(n\in\mathbb N\),
\(\mathbb F\) a field
\(I\) a countable indexing family
[$](X_i,\norm{\ }_{X_i})_{i\in I} [/$] \(\mathbb F\)-normed vector spaces
\(p\ge 1\)
Notation: \(\ell_p\)
Kind: Ex
Index: 7
+: Pointwise sum
0: \((0_i)_{i\in I}\)
×: Pointwise multiplication
Norm: [$]\vec x\mapsto\parallel x\parallel _p=\sqrt[p]{\left(\sum \norm{x}_i^p\right)}[/$]

Definition: The set of continuous function defined on \([a,b]\)
Page: 44
Chapter: 10
Section: 1.2
Variables: \(a\le b\in\mathbb R\)
Notation: \(C[a,b]\)
Index: 8
Kind: Ex

Definition: [$]\max \abs{f(x)}[/$]
Page: 44
Chapter: 10
Section: 1.2
Variables: \(a\le b\in\mathbb R\)
\(f\in C[a,b]\)
Notation: [$]\norm{f}[/$]
Index: 8
Kind: Ex

Definition: [$]\left(\int_a^b\abs{f}^p(x)\dd x\right)^{\frac 1p}[/$]
Page: 44
Chapter: 10
Section: 1.2
Variables: \(a\le b\in\mathbb R\)
\(f\in C[a,b]\),
\(p\ge1\)
Notation: [$]\norm{f}_p[/$]
Index: 8
Kind: Ex

Name: An Hermitian form
Definition: \(f(x,y)=\overline{f(y,x)}\)
Page: 45
Chapter: 10
Section: 1.3
Variables: \(\mathbb F\) a field
\(X\) a vector space
Notation: \(\langle\cdot,\cdot\rangle\)
Definition2: \(f(\lambda x,y)=\lambda f(x,y)\)
Example: [$]\defFun{X\times X}{\mathbb C}{(x,y)}{0}[/$]
Conjdef: A function \(f:X\times X\to\mathbb F\) such that
Definition3: \(f(x+x',y)=f(x,y)+f(x',y)\)
Example2: dot product

Name: A nonnegative Hermitian form
Definition: An hermitian form such that \(\langle x,x\rangle\ge0\)
Page: 45
Chapter: 10
Section: 1.3
Variables: \(\mathbb F\) a field
\(X\) a vector space

Name: A nondegenerate Hermitian form
Definition: An hermitian form such that \(\langle x,x\rangle=0\) iff \(x=0\)
Page: 45
Chapter: 10
Section: 1.3
Variables: \(\mathbb F\) a field
\(X\) a vector space

Name: An inner product in \(X\)
Definition: A nondegenerate nonnegative Hermitian form
Page: 45
Chapter: 10
Section: 1.3
Variables: \(\mathbb F\) a field
\(X\) a vector space
Index: 6, 10, 10, 12
Kind: Def, Ex
Example: [$]\defFun{\mathbb C^n}{\mathbb C}{(x^1,\dots,x^n),(y^1,\dots,y^n)}{\sum x^i\overline{y^i}}[/$]
Example2: [$]\defFun{\ell_2}{\mathbb C}{(x^i)_{i\in\mathbb N},(y^i)_{i\in\mathbb N}}{\sum x^i\overline{y^i}}[/$]
Example3: [$]\defFun{C[a,b]}{\mathbb C}{(f,g)}{\int_a^b(f\cdot\overline g)(x)\dd x}[/$]

Smaller0: [$]\abs{\langle x,y\rangle}^2[/$]
Greater0: \(\langle x,x\rangle\cdot \langle y,y\rangle\)
Page: 46e
Chapter: 10
Section: 1.3
Variables: \(\mathbb F\) a field,
\(X\) a vector space
\(\langle\cdot,\cdot\rangle\) an inner product
\(x,y\in X\)
Smaller to greater: \(\le\)
Name: Cauchy–Bunyakovskii inequality
Greater to smaller: \(\ge\)

Name: The norm associated to \(\langle\cdot,\cdot\rangle\)
Definition: [$]\defFun{X}{\mathbb F}x{\sqrt{\langle x,x\rangle}}[/$]
Page: 47
Chapter: 10
Section: 1.3
Variables: \(\mathbb F\) a field,
\(X\) a vector space
\(\langle\cdot,\cdot\rangle\) an inner product
Denoted by: \(\parallel\cdot \parallel\)

Name: The metric associated to \(\langle\cdot,\cdot\rangle\)
Definition: [$]\defFun{X\times X}{\mathbb F}{(x,y)}{\parallel x-y\parallel}[/$]
Page: 47
Chapter: 10
Section: 1.3
Variables: \(\mathbb F\) a field,
\(X\) a vector space
\(\langle\cdot,\cdot\rangle\) an inner product
Denoted by: \(d\)

Name: An Hermitian space
Definition: finite-dimensional
Page: 48
Chapter: 10
Section: 1.3
Definition2: complex
Conjdef: A vector space
Definition3: with an inner product

Name: An Hilbert space
Definition: whose distance is complete.
Page: 48, https://en.wikipedia.org/wiki/Hilbert_space
Chapter: 10
Section: 1.3
Example: \(L^2(I)\) for \(I\)
Conjdef: A pre-hilbert space
Example2: \(\mathbb R^n\)

Name: A pre-Hilbert space
Definition: with an inner product
Page: 48
Chapter: 10
Section: 1.3
Definition2: over real or complex
Conjdef: A vector space
Extra: (This page ambiguously require the space to be infinite-dimensional. It is probably false but the web does not seems to have any expectation about the dimension)

Name: A linear maping
Definition: \(A(x+x')=A(x)+A(x')\) for \(x,x'\in X\)
Page: 49, 50
Chapter: 10
Section: 2
Variables: \(\mathbb F\) a field
\(X,Y\) vector spaces
Index: 1, 1-4
Kind: Def, Ex
Definition2: \(A(\lambda x)=\lambda A(x)\) for \(\lambda \in\mathbb F\) and \(x\in X\)
Example: [$]\defFun{\substack{\ell\\0}}{\substack{\ell\\0}}{(x_i)_i}{(ix_i)_i}[/$]
Conjdef: A function \(A:X\to Y\) such that
Name2: A linear transformation
Example2: [$]\defFun{C([a,b],\mathbb R)}{\mathbb R}{f}{f(x_0)}[/$] for some \(a\le x_0\le b\) fixed
Example3: [$]\defFun{C([a,b],\mathbb R)}{\mathbb R}{f}{\int_a^b f(x)\dd x}[/$] for some \(a\le b\) fixed
Example4: [$]\defFun{C([a,b],\mathbb R)}{C([a,b],\mathbb R)}{f}{\int_a^x f(t)\dd t}[/$] for some \(a\le b\) fixed

Name: An \(n\)-linear mapping
Definition: is linear in each of its component.
Page: 49, 51
Chapter: 10
Section: 2
Variables: \(\mathbb F\) a field
\(n\in\mathbb N\)
Index: 2, 9
Kind: Def, Ex
Definition2: of the form \((x_1,\dots,x_n)\mapsto \sum A_i(x_i)\) with \(A_i\in\mathcal L(X^i,Y)\)
Conjdef: A function \(A:\Pi_{i=1}^n X_i\to Y\), equivalently

Name: A multilinear mapping
Definition: An \(n\)-linear function for some \(n\).
Page: 49, 50
Chapter: 10
Section: 2
Variables: \(\mathbb F\) a field
\(n\in\mathbb N\)
Kind: Def, Ex
Example: \((x_1,\dots,x_n)\mapsto \prod x_i\) is \(n\)-linear
Example2: inner product in a Euclidean vector space is bilinear
Example3: Cross product in \(E^3\)
Example4: Determinant

Name: The vector space of continuous \(n\)-linear functions from \(\Pi X_i\) to \(Y\)
Page: 49, 57
Chapter: 10
Section: 2
Variables: \(\mathbb F\) a field,
\(n\in\mathbb N\),
\(X_1,\dots,X_n,Y\) vector spaces
Notation: \(\mathcal L(X_1,\dots,X_n;Y)\)
Kind: Def
Typ: A \(\mathbb F\) vector space

Name: A bilinear function
Definition: A 2-linear function
Page: 49
Chapter: 10
Section: 2
Variables: \(\mathbb F\) a field
Kind: Def

Name: A trilinear function
Definition: A 3-linear function
Page: 49
Chapter: 10
Section: 2
Variables: \(\mathbb F\) a field
Kind: Def

Name: A (multi)linear functional
Definition: A (multi)linear mapping where the output is of dimension 1
Page: 445, 49, 50
Chapter: 10, 8, 8
Section: 2, 3, 3
Variables: \(\mathbb F\) a field
Index: ,, 5
Kind: ,, ex
Definition2: A function of the form [$]\defFun{\mathbb F^n}{\mathbb F}{v}{\langle \xi,v\rangle}[/$] for some \(\xi\in\mathbb F^n\) fixed.  (assuming input is \(\mathbb R^n\) as a Euclidean space)
Example: Product of all arguments

Question: Difference between multilinear and \(n\)-linear mapping ?
Answer: To multiply output by \(\lambda\) in an \(n\)-linear/multilinear mapping, you must multiply each input by \(\sqrt[n]\lambda\)/\(\lambda\)
Page: 49
Chapter: 10
Section: 2.1

Name: (multi)linear functionals over \(\prod X_i\)
Definition: (multi)linear mappings \(\prod X_i\to\mathbb F\)
Page: 49
Chapter: 10
Section: 2.1
Variables: \(\mathbb F\)
\(X_1,\dots,X_n\) \(\mathbb F\)-vector space

Name: Linear operator
Definition: Linear mappings with the same domain and co-domain
Page: 49
Chapter: 10
Section: 2.1

Name: multilinear operator
Definition: multilinear mappings with the same domain and co-domain
Page: 49
Chapter: 10
Section: 2.1

Name: The norm of \(A\)
Definition: [$]\sup_{x_1\ne0,\dots,x_n\ne0}\frac{\abs{A(x_1,\dots,x_n)}_Y}{\prod\abs{i_i}_{X_i}}[/$]
Page: 52, 53
Chapter: 10
Section: 2.2
Variables: \(A:X_1\times\dots\times X_n\to Y\) multilinear transformation
Notation: [$]\norm{A}[/$]
Definition2: [$]\sup_{e_1,\dots,e_n}\abs{A(e_1,\dots,e_n)}[/$] with each \(e_i\) of norm \(1\)
Definition3: the least \(M\ge0\) such that [$]\abs{A(x_1,\dots,x_m)}\le M\prod\abs{x_i}[/$]
Intuition: The largest of the semiaxes of the ellipsoid of the image of \(A\) on the sphere of radius 1 in \(X_1\times\dots\times X_n\)

Definition: [$]\norm{x_i}_{X_i}[/$]
Page: 52
Chapter: 10
Section: 2.2
Variables: \(X_1,\dots, X_n\) normed vector spaces
\(x_i\in X_i\)
Context: Vector in space used to create more complex normed vector space
Notation: [$]\abs{x_i}[/$]

Smaller0: [$]\abs{A(x_1,\dots,x_n)}[/$]
Greater0: [$]\norm{A}\prod \abs{x_i}[/$]
Page: 52
Chapter: 10
Section: 2.2
Variables: \(X_1,\dots, X_n\) normed vector spaces
\(x_i\in X_i\)
\(A:X_1\times\dots\times X_n\to Y\) multilinear transformation
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)
Context: Vector in space used to create more complex normed vector space

Name: A bounded multilinear transformation
Definition: whose norm is finite
Page: 53, 56
Chapter: 10
Section: 2.2
Variables: \(X_1,\dots, X_n,Y\) normed vector spaces
\(x_i\in X_i\)
\(A:X_1\times\dots\times X_n\to Y\) multilinear transformation
Index: 4, 1
Kind: Def, Prop
Definition2: continuous
Conjdef: A continuous multilinear, equivalently
Definition3: continuous at an arbitrary point

Smaller0: each \(X_i\) are finite dimensional
Greater0: \(A\) is a bounded multilinear transformation
Page: 53
Chapter: 10
Section: 2.2
Variables: \(X_1,\dots, X_n,Y\) normed vector spaces
\(x_i\in X_i\)
\(A:X_1\times\dots\times X_n\to Y\) multilinear transformation

Question: The norm of [$]\defFun{\substack{\ell\\0}}{\substack{\ell\\0}}{(x_1,\dots, x_n,\dots)}{ (1x_1,\dots,nx_n,\dots)}[/$] is
Answer: \(\infty\). Indeed [$]\norm{e_n}=1[/$] and [$]\norm{Ae_n}=\norm{ne_n}=n[/$]
Page: 53
Chapter: 10
Section: 2.2
Context: Take the norm of a sequence to be the sum of its elements
Index: 1'
Kind: ex

Question: The norm of  [$]\defFun{C[a,b]}{\mathbb R}{f}{f(x_0)}[/$]
Answer: is \(1\)
Page: 53
Chapter: 10
Section: 2.2
Variables: \(a\le x_0\le b\)
Context: Let the norm on \(C[a,b]\) be [$]f\mapsto \max_{a\le x\le b}f(x)[/$]
Index: 2'
Kind: ex
Construction: if \(|f|\le 1\) then \(|Af|=|f(x_0)|\le 1\) and \(|Af|=1\) for \(f=x\mapsto 1\).

Question: The norm of  [$]\defFun{C[a,b]}{\mathbb R}{f}{f(x_0)}[/$]
Answer: is \(\infty\)
Page: 53
Chapter: 10
Section: 2.2
Context: Let the norm on \(C[a,b]\) be [$]f\mapsto\int_a^b\abs{f}(x)\dd x[/$].
\(x_0\in[a,b]\) fixed
Index: 2'
Kind: ex
Construction: take \([a,b]=[0,1]\), \(x_0=1\) and the family of monomial \(f_n=(n+1)x^n\). All of those monomials have norm 1, and [$]\abs{Af_n}=n+1[/$]

Question: The norm of [$]\defFun{C[a,b]}{\mathbb R}{f}{\int_a^bf(x)\dd x}[/$] is
Answer: \(b-a\)
Page: 53
Chapter: 10
Section: 2.2
Context: Let the norm on \(C[a,b]\) be [$]f\mapsto \max_{a\le x\le b}\abs{f(x)}[/$].
Construction: the inequality is straightforward, by integrating a function whose range is in \(B(0,-1)\).
The bound is obtained for the constant function 1

Question: The norm of [$]\defFun{C[a,b]}{C[a,b]}{f}{\defFun{[a,b]}{\mathbb R}{x}{\int_a^xf(t)\dd t}}[/$] is
Answer: \(b-a\)
Page: 53
Chapter: 10
Section: 2.2
Context: Let the norm on \(C[a,b]\) be [$]f\mapsto \max_{a\le x\le b}\abs{f(x)}[/$].
Index: 4'
Kind: ex
Construction: the inequality is straightforward, by integrating a function whose range is in \(B(0,-1)\).
The bound is obtained for the constant function 1

Question: The norm of [$]\defFun{\mathbb R^n}{\mathbb R}{x_1,\dots,x_n}{\prod x_i}[/$] is
Answer: \(1\)
Page: 53
Chapter: 10
Section: 2.2
Index: 5'
Kind: ex

Question: The norm of [$]\defFun{\mathbb R^n,\mathbb R^n}{\mathbb R}{\vec x,\vec y}{\langle\vec x,\vec y\rangle}[/$] is
Answer: \(1\)
Page: 53
Chapter: 10
Section: 2.2
Index: 6'
Kind: ex
Construction: Inequality by Cauchy-Bunyakovskii inequality. Equality thanks to \(\vec x=\vec y\) of norm 1

Question: The norm of [$]\defFun{\mathbb R^3,\mathbb R^3}{\mathbb R^3}{\vec x,\vec y}{[\vec x,\vec y]}[/$] (the cross product) is
Answer: \(1\)
Page: 53
Chapter: 10
Section: 2.2
Index: 7'
Kind: ex
Construction: inequality because [$]\abs{[\vec x,\vec y]}=\abs{\vec x}\abs{\vec y}\sin\phi\le\abs{\vec x}\abs{\vec y}[/$] with \(\phi\) the angle between vectors.
Equality obtained for \(x_1=x_2\) unit vectors

Question: The norm of the determinant function (as a multilinear of \(\left(\mathbb R^n\right)^n\) to \(\mathbb R\)) is
Answer: \(1\)
Page: 53
Chapter: 10
Section: 2.2
Index: 8'
Kind: ex

Question: The norm of the multilinear map [$]\defFun{\prod X_i}{Y}{(x_1,\dots,x_m)}{\sum A_i x_i}[/$] is
Answer: between [$]\max_i\norm{A_i}[/$] and [$]\sum_{i}\norm{A_i}[/$]
Page: 55
Chapter: 10
Section: 2.2
Variables: \(X_1,\dots,X_m,Y\) normed vector spaces
\(A_i:\prod X_i\to Y\) linear
Index: 9'
Kind: ex

Question: The norm of the multilinear map [$]\defFun{X}{\prod Y_i}{x}{(A_i x)}[/$] is
Answer: between [$]\max_i\norm{A_i}[/$] and [$]\sum_{i}\norm{A_i}[/$]
Page: 55
Chapter: 10
Section: 2.2
Variables: \(X,Y_1,\dots, Y_n\) normed vector spaces
\(A_i:\prod X\to Y_i\) linear
Index: 10'
Kind: ex
Extra: There probably miss a norm over the codomain space

Question: The norm of the multilinear map [$]A=\defFun{\prod X_i}{\prod Y_i}{(x_1,\dots,x_m)}{(\sum_{i}A_{i,1} x_i,\dots,\sum_{i}A_{i,n} x_i)}[/$] is
Answer: between [$]\max_{i,j}\norm{A_{i,j}}[/$] and [$]\sum_{i,j}\norm{A_{i,j}}[/$]
Page: 55
Chapter: 10
Section: 2.3
Variables: \(X_1,\dots,X_m,Y_1,\dots, Y_n\) normed vector spaces
\(A_{i,j}:X_i\to Y_j\) linear
Index: 11'
Kind: ex
Extra: There probably miss a norm over the codomain space
Construction: [$]\norm{A}=\sup_{x_1,\dots,x_n,x_i\ne0}\frac{\abs{A(x_1,\dots,x_n)}_Y}{\prod_{i=1}^n\abs{x_i}_{X_i}}[/$]

Name: Complete normed (Banach) space
Page: 58
https://en.wikipedia.org/wiki/Banach_space
Section: 2.3
Chapter: 10
Index: 3
Kind: Prop
Under: \(Y\mapsto\mathcal L(X_1,\dots,X_n;Y)\)
Under2: taking quotient with [$]\norm{x+M}\to\inf_{m\in M}\norm{x+m}[/$]
Condition2: by a closed linear subspace \(M\)
Under3: product for sum norm
Condition3: up to countable
Under4: product for max norm
Condition4: any arity

Smaller0: \(Y\) complete
Greater0: equals [$]\lim\norm{A_i}[/$]
Page: 59
Chapter: 10
Section: 2.3
Variables: \(X_1,\dots,X_n,Y\) normed spaces
\((A_i)_{i\in\mathbb N}\in\mathcal L(X_1,\dots,X_n;Y)\)
Index: 3
Kind: Prop
Prefix Greater: [$]\norm{\lim A_i}[/$]
Smaller1: \((A_i)\) Cauchy
Counterexample: Without completeness, [$]\norm{\lim A_i}[/$] is not even defined

Question: \(\mathcal L\left(X_1,\dots,X_m;\mathcal L(X_{m+1},\dots,X_n;Y)\right)\) is
Answer: in bijection with \(\mathcal L(X_1,\dots,X_n;Y)\), preserving norm and vector-space structure
Page: 59
Chapter: 10
Section: 2.3
Variables: \(X_1,\dots,X_n,Y\) normed spaces
\(0\le m\le n\)
\((A_i)_{i\in\mathbb N}\in\mathcal L(X_1,\dots,X_n;Y)\)
Index: 3
Kind: Prop
Construction: [$]\defFun{\mathcal L\left(X_1,\dots,X_m;\mathcal L(X_{m+1},\dots,X_n;Y)\right)}{\mathcal L(X_1,\dots,X_n;Y)}{A}{\defFun{X_{m+1}\times\dots\times X_n}{Y}{(x_{m+1},\dots,x_n)}{A(x_1,\dots,x_m)(x_{m+1},\dots,x_n)}}[/$].

Question: Example of two norms such that the identity on \(C([0,1],\mathbb R)\) is not continuous.
Answer: In the domain, the norm is [$]f\mapsto\int_a^b\abs{f(x)}[/$].
In the range, the norm is [$]f\mapsto\max_x\abs{f(x)}[/$].
Take the growing steeple sequence
\(f_n(x)=\left\{\begin{array}{ll}n^2x & \text{if }0\le x\le \frac 1n\\2n-n^2x & \text{if }\frac 1n\le x\le\frac2n\\0 & \text{if }\frac2n\le x\le 1\end{array}\right.\)
This sequence of norm is the constant 1 in the domain, and is the sequence \((n)\) in the range
Page: 60
Chapter: 10
Section: 2.4
Index: 2.C
Kind: Ex

Page: 43
Chapter: 10
Section: 1.2
Variables: \(n\ge 1\)
\(n\in\mathbb N\)
Notation: \(\mathbb R^n_p\)
Kind: Ex
Index: 5
Norm: [$]\norm{\vec x}_p=\sqrt[p]{\sum x_i^p}[/$]
Context: Analysis