Title: Theory / Unimportant / Math / Analysis / Functional - Haïm Brezis
Tags: récupéré fromMath hasSeenConjDef closed defToAlgebra mathToSee setNorm

Name: A functional
Definition: A function, from some subspace of \(E\), to \(\mathbb R\)
Page: 1
Variables: \(E\) a real vector space

Name: A Minkowski functional
Definition: homogeneous
Page: 1
Variables: \(E\) a real vector space
Definition2: subadditive
Conjdef: A functional

Name: An inductive ordered set
Definition: An ordered set whose every totally ordered subset has an upper bound
Page: 1

Smaller0: \(p\) is Minkowski on \(G\)
Greater0: \(p\) admits a Minkowski extension on \(E\)
Page: 1
Variables: \(E\) a real vector space
\(p:E\to\mathbb R\)
\(G\subseteq E\) a subspace
Smaller to greater: iff
Assuming: Zorn's lemma
Name: Hahn-Banach analytic form

Smaller0: \(P\) is inductive
Greater0: \(P\) has a maximal element
Page: 2
Chapter: 1
Variables: \((P,\le)\) an ordered set
Smaller to greater: iff
Name: Zorn's lemma
Extra: duplicatenin non standard

Name: normed vector space
Page: 2
Chapter: 1
Abbreviation: n.v.s

Page: 2
Chapter: 1
Name: The dual space of \(E\)
Set: The set continuous linear functional
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
Notation: \(E^\star\)
0: \(\cdot\mapsto 0\)
Typ: The n.v.s.
Norm: [$]f\mapsto\text{sup}_{\abs{x}\le 1, x\in E}f(x)[/$]

Name: The scalar product for the duality \(E^\star,E\)
Definition: \(f(x)\)
Page: 3
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(f\in E^\star\)
\(x\in E\)
Notation: \(\langle f,x\rangle\)

Name: A strictly convex vector space
Definition: A normed vector space with [$]\norm{tx+(1-t)y}<1[/$] for each \(t\in (0,1)\), \(x,y\) distinct of norm 1
Page: 4
Chapter: 1
Index: 2
Kind: Footnote
Example: Hilbert spaces
Example2: \(L^p(\Omega)\) with \(1<p<\infty\)

Definition: [$]\norm{f_0}=\norm{x_0}[/$]
Page: 3, 4
Chapter: 1
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(x_0\in E\)
Index: 3
Kind: Cor
Definition2: [$]\langle f_0,x_0\rangle=\norm{x_0}^2[/$]
Conjdef: The set of \(f_0\in E^\star\) such that
Cardinal: Non-empty

Name: The duality map
Definition: [$]\defFun{E}{E^\star}{x_0}{F(x_0)}[/$]
Page: 4
Chapter: 1
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space

Name: An affine hyperplane of \(E\)
Definition: A set of the form \([f=\alpha]\)
Page: 4
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(x\in E\)
Denoted by: \(H\)
Assuming: \(f\) does not vanish identically

Definition: The set of \(x\in E\) such that \(f(x)=\alpha\)
Page: 4
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(f\) a linear functional
\(\alpha\in\mathbb K\) a constant
Notation: \([f=\alpha]\)

Name: An hyperplane which separates \(A\) and \(B\)
Definition: An hyperplane \([f=\alpha]\) such that \(f(x)\le\alpha\) for each \(x\in A\) and \(f(x)\ge\alpha\) for each \(x\in B\)
Page: 5
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(A,B\subseteq E\)

Name: An hyperplane which strictly separates \(A\) and \(B\)
Definition: An hyperplane \([f=\alpha]\) such that \(f(x)\le\alpha-\epsilon\) for each \(x\in A\) and \(f(x)\ge\alpha+\epsilon\) for each \(x\in B\) and some \(\epsilon>0\)
Page: 5
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(A,B\subseteq E\)
Intuition: \(A\) and \(B\) lies on different sides of the hyperplane

Name: The equation of \(H\)
Definition: The equation \(f=\alpha\) such that \(H=[f=\alpha]\)
Page: 5
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
An hyperplane \(H\)

Smaller0: \(A,B\) convex
Greater0: there exists a closed hyperplane that strictly separates \(A\) and \(B\)
Page: 7
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(A,B\subseteq E\)
Kind: Theo
Smaller1: \(A\) disjont from \(B\)
Name: Hahn–Banach, second geometric form
Connect Smaller0: and either
Smaller2: \(A\) closed
Smaller3: \(B\) compact

Smaller0: \(A,B\) convex
Greater0: there exists a closed hyperplane that separates \(A\) and \(B\)
Page: 5, 8
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(A,B\subseteq E\)
Index: 6
Kind: Theo
Assuming: \(A\) disjoint from \(B\)
Name: Hahn–Banach, first geometric form
Smaller2: \(A\) or \(B\) open
Suffix Smaller: )
Smaller3: \(E\) finite dimensional
Connect Smaller1: and (either
Connect Smaller2: or

Name: The gauge of \(A\)
Definition: [$]\defFun{A}{\overline{\mathbb K}}{x}{\text{inf}\{\alpha>0;\alpha^{-1}x\in A\}}[/$]
Page: 5
Chapter: 1
Section: 2
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(A\subseteq E\)
Notation: \(p\)
Index: 6
Kind: Theo
Name2: The Minkowski function of \(A\)
Typ: Minkowski

Definition: [$]x\mapsto\defFun{E^\star}{E}{f}{\langle f,x\rangle}[/$]
Page: 8
Section: 3
Chapter: 1
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb R\) the application function
Notation: \(J\)
Name: The canonical injection to the bidual
Domain: \(E\)
Codomain: \(E^{\star\star}\)
Linear: Linear
Measure: isometry

Name: A reflexive n.v.s.
Definition: A n.v.s. such that \(J\) is surjective
Page: 9
Chapter: 1
Section: 3
Variables: \(\mathbb K\) a field

Definition: The set of \(f\in E^\star\) such that \(\langle f,x\rangle=0\) for each \(x\in M\)
Page: 9
Chapter: 1
Section: 3
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(M\subseteq E\) linear
\(\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb R\) the application function
Notation: \(M^\bot\)

Name: The space orthogonal to \(M\)
Definition: The set of \(f\in E^\star\) such that \(\langle f,x\rangle=0\) for each \(x\in M\)
Page: 9
Chapter: 1
Section: 3
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(M\subseteq E\) linear
\(\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb R\) the application function
Notation: \(M^\bot\)
Typ: Closed linear subspace of \(E^\star\)

Name: The space orthogonal to \(N\)
Definition: The set of \(x\in E\) such that \(\langle f,x\rangle=0\) for each \(f\in N\)
Page: 9
Chapter: 1
Section: 3
Variables: \(\mathbb K\) a field
[$](E,\norm{})[/$] a normed vector space
\(M\subseteq E^\star\) linear
\(\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb R\) the application function
Notation: \(N^\bot\)
Typ: Closed linear subspace of \(E^\star\)

Name: The domain of \(\phi\)
Definition: the set of \(x\in E\) such that \(\phi(x)\in\mathbb R\)
Page: 10
Chapter: 1
Section: 4
Variables: \(\mathbb K\) a field
\(E\) a topological space
\(\phi:E\to (-\infty,+\infty]\)
Notation: \(D(\phi)\)

Name: The epigraph of \(\phi\)
Definition: the set of \([x,\lambda]\in E\times \mathbb R\) such that \(\phi(x)\le\lambda\)
Page: 10
Chapter: 1
Section: 4
Variables: \(\mathbb K\) a field
\(E\) a topological space
\(\phi:E\to (-\infty,+\infty]\)
Abbreviation: \(\text{epi}(\phi)\)

Definition: \(\{x\in E; \phi(x)\le\lambda\}\)
Page: 10
Chapter: 1
Section: 4
Variables: \(\mathbb K\) a field
\(E\) a topological space
\(\phi:E\to (-\infty,+\infty]\)
Notation: \([\phi\le\lambda]\)

Name: A lower semicontinuous function on \(E\)
Definition: for every \(\lambda\in\mathbb R\), \([\phi\le\lambda]\) is closed
Page: 10
Chapter: 1
Section: 4
Variables: \(\mathbb K\) a field
\(E\) a topological space
Definition2: \(\text{epi}\phi\) is closed
Conjdef: A function \(\phi:E\to (-\infty,+\infty]\) such that, equivalently
Definition3: for every \(x\in E\), \(\epsilon>0\) there is some neighborhood \(V\) of \(x\) sach that for each \(y\in V\),  \(\phi(y)\ge\phi(x)-\epsilon\)
Abbreviation: l.s.c.
Definition4: for each converging sequence \(x_n\), \(\lim\inf_{n\to\infty}\phi(x_n)\ge\phi (\lim_{n\to\infty}x)\) (assuming \(E\) metric)

Name: l.s.c.
Page: 10
Section: 4
Chapter: 1
Variables: \(\mathbb K\) a field
\(E\) a topological space
Under: sum
Under2: superior envelope

Name: The superior envelope of \((\phi_i)\)
Definition: [$]\defFun{E}{\mathbb K}{x}{\sup_{i}\phi_i(x)}[/$]
Page: 10
Chapter: 1
Section: 4
Variables: \(\mathbb K\) an ordered field
\(E\) a topological space
\(\phi_i:E\to\mathbb k\) a family of functions

Name: A convex function with values in \((-\infty,+\infty]\)
Definition: A function \(\phi:E\to(-\infty,+\infty]\) such that \(\phi(tx+(1-t)y)\le t\phi(x)+(1-t)\phi(y)\) for each \(x,y\ E\), \(t\in(0,1)\)
Page: 11
Chapter: 1
Section: 4
Variables: \(\mathbb K\) an ordered field
\(E\) a v.s.

Name: Convex maps with values in \((-\infty,+\infty]\)
Page: 11
Section: 4
Chapter: 1
Variables: \(\mathbb K\) an ordered field
\(E\) a v.s.
Under: sum
Under2: superior envelope

Name: The conjugate function
Definition: [$]\defFun{E^\star}{(-\infty,+\infty]}{f}{\sup_{x\in E}\{\langle f,x\rangle-\phi(x)\}}[/$]
Page: 11
Chapter: 1
Section: 4
Variables: \(\mathbb K\) an ordered field
\(E\) a n.v.s.
\(\phi:E\to(-\infty,+\infty]\)
\(\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb R\) the application function
Notation: \(\phi^\star\)
Example: ?
Name2: The Legendre transform of \(\phi\)
Typ: Convex and l.s.c.
Assuming: \(\phi\ne\infty\)

Question: \(\phi(x)+\phi^\star(f)\) is
Answer: at least \(\langle f,x\rangle\)
Page: 11
Chapter: 1
Section: 4
Variables: \(\mathbb K\) an ordered field
\(E\) a n.v.s.
\(\phi:E\to(-\infty,+\infty]\)
\(\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb R\) the application function
Name: Young's inequality

Name: The decimal of \(x\) in the scale of \(r\)
Definition: A sequence of digits, \((d_i)\in\{0,\dots, r-1\}\) such that \(x=\sum_{i}d_i r^{-i}\)
Page: 144
Chapter: 9
Section: 3
Variables: \(x\in[0,1]\)
\(r\in\mathbb N^{>1}\)

Name: Irationals
Page: 145
Chapter: 9
Section: 4
Index: 137
Kind: Theo
Example: the concatenation of the words \((0\dots 01)_i\), where the \(i\)-th word has \(i\) 0
Example2: the binary words whose \(i\)-th letter is one iff \(i\) is prime, otherwise it is 0
Example3: the concatenation of the representations of the primes