Rich families in Asplund spaces and
separable reduction of Fréchet
(sub)differentiability
Marek Cúth, Marián Fabian
Warwick, June 2015
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Rich families in Asplund spaces and
separable reduction of Fréchet
(sub)differentiability
Marek Cúth, Marián Fabian
Warwick, June 2015
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
(X , k · k) a Banach space, V ⊂ X a subspace.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
(X , k · k) a Banach space, V ⊂ X a subspace.
Question. ?Is V ∗ ⊂ X ∗ ?
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
(X , k · k) a Banach space, V ⊂ X a subspace.
Question. ?Is V ∗ ⊂ X ∗ (up to an isometry)?
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
(X , k · k) a Banach space, V ⊂ X a subspace.
Question. ?Is V ∗ ⊂ X ∗ (up to an isometry)?
Why is the question interesting?
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
(X , k · k) a Banach space, V ⊂ X a subspace.
Question. ?Is V ∗ ⊂ X ∗ (up to an isometry)?
Why is the question interesting?
S
If so, that is V ∗ ≡ Y ⊂ X ∗ ,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
(X , k · k) a Banach space, V ⊂ X a subspace.
Question. ?Is V ∗ ⊂ X ∗ (up to an isometry)?
Why is the question interesting?
S
If so, that is V ∗ ≡ Y ⊂ X ∗ , then the mapping
X ∗ 3 x ∗ 7−→ S x ∗ |V ) =: Px ∗
generates a linear norm-one projection on X ∗ , with PX ∗ = Y .
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
(X , k · k) a Banach space, V ⊂ X a subspace.
Question. ?Is V ∗ ⊂ X ∗ (up to an isometry)?
Why is the question interesting?
S
If so, that is V ∗ ≡ Y ⊂ X ∗ , then the mapping
X ∗ 3 x ∗ 7−→ S x ∗ |V ) =: Px ∗
generates a linear norm-one projection on X ∗ , with PX ∗ = Y .
Examples
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 1 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 1 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 1 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 1 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 1 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
• R is σ-complete,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 1 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
•SR is σ-complete, that is, if U1 ⊂ U2 ⊂ · · · is a sequence in R, then
Ui ∈ R.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 1 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
•SR is σ-complete, that is, if U1 ⊂ U2 ⊂ · · · is a sequence in R, then
Ui ∈ R.
Proposition 2
(Important) If R1 , R2 are rich families, then R1 ∩ R2 is rich.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 3 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
•SR is σ-complete, that is, if U1 ⊂ U2 ⊂ · · · is a sequence in R, then
Ui ∈ R.
Proposition 4
(Important) If R1 , R2 , . . . are rich families, then R1 ∩ R2 ∩ · · · is rich.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 3 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
•SR is σ-complete, that is, if U1 ⊂ U2 ⊂ · · · is a sequence in R, then
Ui ∈ R.
Proposition 4
(Important) If R1 , R2 , . . . are rich families, then R1 ∩ R2 ∩ · · · is rich.
Joke. Given a Banach space X , by a rectangle
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 3 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
•SR is σ-complete, that is, if U1 ⊂ U2 ⊂ · · · is a sequence in R, then
Ui ∈ R.
Proposition 4
(Important) If R1 , R2 , . . . are rich families, then R1 ∩ R2 ∩ · · · is rich.
Joke. Given a Banach space X , by a rectangle we understand any
product V × Y where V ∈ S(X ) and Y ∈ S(X ∗ ).
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 3 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
•SR is σ-complete, that is, if U1 ⊂ U2 ⊂ · · · is a sequence in R, then
Ui ∈ R.
Proposition 4
(Important) If R1 , R2 , . . . are rich families, then R1 ∩ R2 ∩ · · · is rich.
Joke. Given a Banach space X , by a rectangle we understand any
product V × Y where V ∈ S(X ) and Y ∈ S(X ∗ ).
Let S<= (X × X ∗ ) denote the family of all such rectangles.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Definition 3 (Borwein-Moors 00)
Let E be a Banach space and let S(E) denote the family of all closed
linear separable subspaces. A subfamily R ⊂ S(E) is called rich if
• R is cofinal, that is, for every Z ∈ S(E) there is U ∈ R so that
U ⊃ Z , and
•SR is σ-complete, that is, if U1 ⊂ U2 ⊂ · · · is a sequence in R, then
Ui ∈ R.
Proposition 4
(Important) If R1 , R2 , . . . are rich families, then R1 ∩ R2 ∩ · · · is rich.
Joke. Given a Banach space X , by a rectangle we understand any
product V × Y where V ∈ S(X ) and Y ∈ S(X ∗ ).
Let S<= (X × X ∗ ) denote the family of all such rectangles.
Clearly, S<= (X × X ∗ ) is a rich subfamily of S(X × X ∗ ).
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry;
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ C the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ C the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is
surjective.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ C the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is
surjective.
Proof.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ C the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is
surjective.
Proof. (ii)⇒(iii) is trivial and (iii)⇒(i) is very easy.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ C the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is
surjective.
Proof. (ii)⇒(iii) is trivial and (iii)⇒(i) is very easy.
(i)⇒(ii) profits from a long bow of ideas across half a century:
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ C the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is
surjective.
Proof. (ii)⇒(iii) is trivial and (iii)⇒(i) is very easy.
(i)⇒(ii) profits from a long bow of ideas across half a century:
[Lindenstrauss65, Amir-Lindenstrauss68, Tacon70, John-Zizler74,
Gul’ko79, Fabian-Godefroy88, Stegall94,Cúth-Fabian15].
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 5
For a Banach space (X , k · k) the following assertions are mutually
equivalent:
(i) X is Asplund, that is, for every Z ∈ S(X ) the dual Z ∗ is separable.
(ii) There exists a rich family A ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ A the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗
is a surjective isometry; moreover, the “projection”
πX (A) := V ∈ S(X ) : V × Y ∈ A for some Y ∈ S(X ∗ )
is a rich family in S(X ).
(iii) There exists a cofinal family C ⊂ S<= (X × X ∗ ) such that
for every V × Y ∈ C the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is
surjective.
Proof. (ii)⇒(iii) is trivial and (iii)⇒(i) is very easy.
(i)⇒(ii) profits from a long bow of ideas across half a century:
[Lindenstrauss65, Amir-Lindenstrauss68, Tacon70, John-Zizler74,
Gul’ko79, Fabian-Godefroy88, Stegall94,Cúth-Fabian15].
Even if the norm k · k on X is Fréchet smooth, a non-negligible work is
needed.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Sketch how to produce the rich family A ⊂ S<= (X × X ∗ ) from the
existence of a function f : X −→ [0, +∞] which is C 1 -smooth on
dom f ,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Sketch how to produce the rich family A ⊂ S<= (X × X ∗ ) from the
existence of a function f : X −→ [0, +∞] which is C 1 -smooth on
dom f , and satisfies that f 0 (V )|V is dense in V ∗ for every V ∈ S(X ).
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Sketch how to produce the rich family A ⊂ S<= (X × X ∗ ) from the
existence of a function f : X −→ [0, +∞] which is C 1 -smooth on
dom f , and satisfies that f 0 (V )|V is dense in V ∗ for every V ∈ S(X ).
(Such an f exists if the norm k · k on X is Frećhet smooth.)
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Sketch how to produce the rich family A ⊂ S<= (X × X ∗ ) from the
existence of a function f : X −→ [0, +∞] which is C 1 -smooth on
dom f , and satisfies that f 0 (V )|V is dense in V ∗ for every V ∈ S(X ).
(Such an f exists if the norm k · k on X is Frećhet smooth.)
The candidate for A may look as:
the family consisting of all rectangles
sp C × sp f 0 (spQ C),
Marek Cúth, Marián Fabian
C ⊂ X countable
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Sketch how to produce the rich family A ⊂ S<= (X × X ∗ ) from the
existence of a function f : X −→ [0, +∞] which is C 1 -smooth on
dom f , and satisfies that f 0 (V )|V is dense in V ∗ for every V ∈ S(X ).
(Such an f exists if the norm k · k on X is Frećhet smooth.)
The candidate for A may look as:
the family consisting of all rectangles
sp C × sp f 0 (spQ C),
C ⊂ X countable
which moreover satisfy
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Sketch how to produce the rich family A ⊂ S<= (X × X ∗ ) from the
existence of a function f : X −→ [0, +∞] which is C 1 -smooth on
dom f , and satisfies that f 0 (V )|V is dense in V ∗ for every V ∈ S(X ).
(Such an f exists if the norm k · k on X is Frećhet smooth.)
The candidate for A may look as:
the family consisting of all rectangles
sp C × sp f 0 (spQ C),
C ⊂ X countable
which moreover satisfy
∗
sp f 0 (spQ C) 3 x ∗ 7−→ x ∗ |sp C ∈ sp C .
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
APPLICATION FOR FRÉCHET SUBDIFFERENTIABILITY
“Definition”. Fréchet subdifferentiability means the lower/bottom part
of Fréchet differentiability.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
APPLICATION FOR FRÉCHET SUBDIFFERENTIABILITY
“Definition”. Fréchet subdifferentiability means the lower/bottom part
of Fréchet differentiability. More exactly
Let (X , k · k) be a Banach space, let f : X −→ (−∞, +∞] be any
proper function, i.e. f 6≡ +∞, and let x ∈ dom f .
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
APPLICATION FOR FRÉCHET SUBDIFFERENTIABILITY
“Definition”. Fréchet subdifferentiability means the lower/bottom part
of Fréchet differentiability. More exactly
Let (X , k · k) be a Banach space, let f : X −→ (−∞, +∞] be any
proper function, i.e. f 6≡ +∞, and let x ∈ dom f .
The Fréchet subdifferential ∂f (x) of f at x is the (possibly empty) set
consisting of all x ∗ ∈ X ∗ such that
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
APPLICATION FOR FRÉCHET SUBDIFFERENTIABILITY
“Definition”. Fréchet subdifferentiability means the lower/bottom part
of Fréchet differentiability. More exactly
Let (X , k · k) be a Banach space, let f : X −→ (−∞, +∞] be any
proper function, i.e. f 6≡ +∞, and let x ∈ dom f .
The Fréchet subdifferential ∂f (x) of f at x is the (possibly empty) set
consisting of all x ∗ ∈ X ∗ such that
f (x + h) − f (x) − hx ∗ , hi > −o(khk) for all
Marek Cúth, Marián Fabian
0 6= h ∈ X
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
APPLICATION FOR FRÉCHET SUBDIFFERENTIABILITY
“Definition”. Fréchet subdifferentiability means the lower/bottom part
of Fréchet differentiability. More exactly
Let (X , k · k) be a Banach space, let f : X −→ (−∞, +∞] be any
proper function, i.e. f 6≡ +∞, and let x ∈ dom f .
The Fréchet subdifferential ∂f (x) of f at x is the (possibly empty) set
consisting of all x ∗ ∈ X ∗ such that
f (x + h) − f (x) − hx ∗ , hi > −o(khk) for all
where o : (0, +∞) −→ [0, +∞] statisfies that
Marek Cúth, Marián Fabian
o(t)
t
0 6= h ∈ X
→ 0 as t ↓ 0.
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
APPLICATION FOR FRÉCHET SUBDIFFERENTIABILITY
“Definition”. Fréchet subdifferentiability means the lower/bottom part
of Fréchet differentiability. More exactly
Let (X , k · k) be a Banach space, let f : X −→ (−∞, +∞] be any
proper function, i.e. f 6≡ +∞, and let x ∈ dom f .
The Fréchet subdifferential ∂f (x) of f at x is the (possibly empty) set
consisting of all x ∗ ∈ X ∗ such that
f (x + h) − f (x) − hx ∗ , hi > −o(khk) for all
where o : (0, +∞) −→ [0, +∞] statisfies that
o(t)
t
0 6= h ∈ X
→ 0 as t ↓ 0.
Easy fact.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
APPLICATION FOR FRÉCHET SUBDIFFERENTIABILITY
“Definition”. Fréchet subdifferentiability means the lower/bottom part
of Fréchet differentiability. More exactly
Let (X , k · k) be a Banach space, let f : X −→ (−∞, +∞] be any
proper function, i.e. f 6≡ +∞, and let x ∈ dom f .
The Fréchet subdifferential ∂f (x) of f at x is the (possibly empty) set
consisting of all x ∗ ∈ X ∗ such that
f (x + h) − f (x) − hx ∗ , hi > −o(khk) for all
where o : (0, +∞) −→ [0, +∞] statisfies that
o(t)
t
0 6= h ∈ X
→ 0 as t ↓ 0.
Easy fact. If ∂f (x) 6= ∅ and also ∂(−f )(x) 6= ∅, then f is Fréchet
differentiable at x.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x),
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
and for every x ∈ V ,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then
∃ x ∗ ∈ ∂f (x),
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then
∃ x ∗ ∈ ∂f (x), even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ ,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then
∃ x ∗ ∈ ∂f (x), even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ , even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ and that kx ∗ k = kv ∗ k,
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then
∃ x ∗ ∈ ∂f (x), even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ , even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ and that kx ∗ k = kv ∗ k, even
∃ !x ∗ ∈ ∂f (x) ∩ Y such that x ∗ |V = v ∗ and that kx ∗ k = kv ∗ k;
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then
∃ x ∗ ∈ ∂f (x), even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ , even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ and that kx ∗ k = kv ∗ k, even
∃ !x ∗ ∈ ∂f (x) ∩ Y such that x ∗ |V = v ∗ and that kx ∗ k = kv ∗ k;
Summarizing:
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Theorem 6
(Fabian-Živkov85, Fabian 89, Fabian-Ioffe15) Let (X , k · k) be any
Banach space and let f : X −→ (−∞, +∞] be any proper function,
i.e. f 6≡ +∞. Then there exists a cofinal, even rich, family R ⊂ S(X )
such that for every V ∈ R and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then ∃ x ∗ ∈ ∂f (x), i.e. ∂(f |V )(x) 6= ∅ ⇒ ∂f (x) 6= ∅.
Theorem 7
(Main, Cúth-Fabian15) Let (X , k · k) be any Asplund space and
f : X −→ (−∞, +∞] any proper function. Then there exists a rich
family R ⊂ A ⊂ S<= (X × X ∗ ) such that for every V × Y ∈ R
the assignment Y 3 x ∗ 7−→ x ∗ |V ∈ V ∗ is a surjective isometry
and for every x ∈ V ,
if ∃ v ∗ ∈ ∂(f |V )(x), then
∃ x ∗ ∈ ∂f (x), even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ , even
∃ x ∗ ∈ ∂f (x) such that x ∗ |V = v ∗ and that kx ∗ k = kv ∗ k, even
∃ !x ∗ ∈ ∂f (x) ∩ Y such that x ∗ |V = v ∗ and that kx ∗ k = kv ∗ k;
Summarizing: ∂(f |V )(x) = ∂f (x) ∩ Y |V = ∂f (x) |V for every x ∈ V .
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Some applications
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Some applications
(Preiss84, Zajı́ček12) Fréchet differentiability of any function on an
Asplund space is separably reducible via a suitable cofinal, even
rich, family.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Some applications
(Preiss84, Zajı́ček12) Fréchet differentiability of any function on an
Asplund space is separably reducible via a suitable cofinal, even
rich, family.
Proof. Let R1 , R2 be the rich families separably reducing the Fréchet
subdifferentiability of f and −f respectively.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Some applications
(Preiss84, Zajı́ček12) Fréchet differentiability of any function on an
Asplund space is separably reducible via a suitable cofinal, even
rich, family.
Proof. Let R1 , R2 be the rich families separably reducing the Fréchet
subdifferentiability of f and −f respectively. Put then R := R1 ∩ R2 .
This R works.
(Mordukhovič-Fabian02) Non-zeroness of Fréchet cone (∂ιΩ (x) on an
Asplund space is separably reducible via a suitable cofinal, even
rich, family.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
Some applications
(Preiss84, Zajı́ček12) Fréchet differentiability of any function on an
Asplund space is separably reducible via a suitable cofinal, even
rich, family.
Proof. Let R1 , R2 be the rich families separably reducing the Fréchet
subdifferentiability of f and −f respectively. Put then R := R1 ∩ R2 .
This R works.
(Mordukhovič-Fabian02) Non-zeroness of Fréchet cone (∂ιΩ (x) on an
Asplund space is separably reducible via a suitable cofinal, even
rich, family.
(Fabian89, Mordukhovič-Fabian02) Fuzzy calculus for Fréchet
subdifferentials on an Asplund space is separably reducible via a
suitable cofinal, even rich, family.
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
THANK YOU FOR YOUR ATTENTION
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
THANK YOU FOR YOUR ATTENTION
http://arxiv.org/abs/1505.07604
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)
THANK YOU FOR YOUR ATTENTION
http://arxiv.org/abs/1505.07604
Marián Fabian
Mathematical Institute, Czech Academy of Sciences
Žitná 25, 115 67 Praha 1, Czech Republic
fabian@math.cas.cz
Marek Cúth
marek.cuth@gmail.com
Marek Cúth, Marián Fabian
Rich families in Asplund spaces and separable reduction of Fréchet (sub)