A separable Frchet space of almost universal disposition Christian - PowerPoint PPT Presentation

A separable Fréchet space of almost universal disposition Christian Bargetz joint work with Jerzy Kąkol and Wiesław Kubiś University of Innsbruck Paweł Domański Memorial Conference Będlewo 1–7 July 2018

Notation Let E and F be Banach spaces. A linear mapping f : E → F , with � f ( x ) � F = � x � E for all x ∈ E is called an isometric embedding. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 2 / 17

Notation Let E and F be Banach spaces. A linear mapping f : E → F , with � f ( x ) � F = � x � E for all x ∈ E is called an isometric embedding. Given ε > 0, a linear mapping ( 1 + ε ) − 1 � x � E ≤ � f ( x ) � F ≤ ( 1 + ε ) � x � E f : E → F , with for all x ∈ E is called an ε -isometric embedding. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 2 / 17

A universal Banach space Theorem (Banach-Mazur, 1929) The space ( C [ 0 , 1 ] , � · � ∞ ) is universal for all separable Banach spaces. In other words, for every separable Banach space E there is an isometric embedding E ֒ → C [ 0 , 1 ] . Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 3 / 17

The Gurari˘ ı space In 1965, V. I. Gurari˘ ı constructed a separable Banach space with the following extension property. (G) For every ε > 0, for all finite dimensional normed spaces E ⊆ F , for every isometric embedding e : E → G there exists an ε -isometric embedding f : F → G such that f ↾ E = e . Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 4 / 17

The Gurari˘ ı space In 1965, V. I. Gurari˘ ı constructed a separable Banach space with the following extension property. (G) For every ε > 0, for all finite dimensional normed spaces E ⊆ F , for every isometric embedding e : E → G there exists an ε -isometric embedding f : F → G such that f ↾ E = e . In other words, E F e G Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 4 / 17

The Gurari˘ ı space In 1965, V. I. Gurari˘ ı constructed a separable Banach space with the following extension property. (G) For every ε > 0, for all finite dimensional normed spaces E ⊆ F , for every isometric embedding e : E → G there exists an ε -isometric embedding f : F → G such that f ↾ E = e . In other words, E F e f G Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 4 / 17

The Gurari˘ ı space In 1965, V. I. Gurari˘ ı constructed a separable Banach space with the following extension property. (G) For every ε > 0, for all finite dimensional normed spaces E ⊆ F , for every isometric embedding e : E → G there exists an ε -isometric embedding f : F → G such that f ↾ E = e . In other words, E F e f G In 1976, W. Lusky showed that (G) defines G uniquely up to isometry. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 4 / 17

The Gurari˘ ı space In 1965, V. I. Gurari˘ ı constructed a separable Banach space with the following extension property. (G) For every ε > 0, for all finite dimensional normed spaces E ⊆ F , for every isometric embedding e : E → G there exists an ε -isometric embedding f : F → G such that f ↾ E = e . In other words, E F e f G In 1976, W. Lusky showed that (G) defines G uniquely up to isometry. A simpler proof: Kubiś and Solecki (2013) Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 4 / 17

Graded Fréchet spaces We consider Fréchet spaces with a fixed sequence of semi-norms. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 5 / 17

Graded Fréchet spaces We consider Fréchet spaces with a fixed sequence of semi-norms. If in addition, � · � 1 ≤ � · � 2 ≤ . . . we call E a graded Fréchet space. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 5 / 17

( ε -)Isometries Let E and F be Fréchet spaces with fixed sequences of semi-norms. A linear mapping f : E → F , with � f ( x ) � F , i = � x � E , i for all i ∈ N and all x ∈ E is called an isometric embedding. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 6 / 17

( ε -)Isometries Let E and F be Fréchet spaces with fixed sequences of semi-norms. A linear mapping f : E → F , with � f ( x ) � F , i = � x � E , i for all i ∈ N and all x ∈ E is called an isometric embedding. Given ε > 0, a linear mapping ( 1 + ε ) − 1 � x � E , i ≤ � f ( x ) � F , i ≤ ( 1 + ε ) � x � E , i f : E → F , with for all i ∈ N and all x ∈ E is called an ε -isometric embedding. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 6 / 17

A universal Fréchet space Theorem (Mazur-Orlicz, 1948) The space ( C ( R ) , {� · � i } i ∈ N ) , where � � � f � i := sup | f ( x ) | : x ∈ [ − i , i ] , is universal for all separable Fréchet spaces. In other words, for every separable Fréchet space E there is an isometric embedding E ֒ → C ( R ) . Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 7 / 17

Is there a Fréchet-Gurari˘ ı space? Question Is there an analogue of the Gurari˘ ı space for separable Fréchet spaces? Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 8 / 17

Is there a Fréchet-Gurari˘ ı space? Question Is there an analogue of the Gurari˘ ı space for separable Fréchet spaces? A natural candidate is G N . Can we find a suitable sequence of semi-norms on G N ? Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 8 / 17

Connecting to the Banach space setting Let ( E , {� · �} i ∈ N ) be a finite dimensional graded Fréchet space. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 9 / 17

Connecting to the Banach space setting Let ( E , {� · �} i ∈ N ) be a finite dimensional graded Fréchet space. Observations: ker � · � i = { x ∈ E : � x � i = 0 } is a subspace. Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 9 / 17

Connecting to the Banach space setting Let ( E , {� · �} i ∈ N ) be a finite dimensional graded Fréchet space. Observations: ker � · � i = { x ∈ E : � x � i = 0 } is a subspace. � · � i is a norm on E i := E / ker � · � i . Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 9 / 17

Connecting to the Banach space setting Let ( E , {� · �} i ∈ N ) be a finite dimensional graded Fréchet space. Observations: ker � · � i = { x ∈ E : � x � i = 0 } is a subspace. � · � i is a norm on E i := E / ker � · � i . ( E i , � · � i ) is a Banach space and � E ֒ → E i and E i + 1 ։ E i i Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 9 / 17

Connecting to the Banach space setting Let ( E , {� · �} i ∈ N ) be a finite dimensional graded Fréchet space. Observations: ker � · � i = { x ∈ E : � x � i = 0 } is a subspace. � · � i is a norm on E i := E / ker � · � i . ( E i , � · � i ) is a Banach space and � E ֒ → E i and E i + 1 ։ E i i If f : E → F is ( ε -)isometric, so is f i : E i → F i , defined by f E F can can f i E i F i Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 9 / 17

An important tool: A universal operator on G There exists a non-expansive linear operator π : G → G with ker π ≃ G and Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 10 / 17

An important tool: A universal operator on G There exists a non-expansive linear operator π : G → G with ker π ≃ G and 1 For every separable Banach space X and � T � ≤ 1, π G G T X Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 10 / 17

An important tool: A universal operator on G There exists a non-expansive linear operator π : G → G with ker π ≃ G and 1 For every separable Banach space X and � T � ≤ 1, ∃ i π G G T i X Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 10 / 17

An important tool: A universal operator on G There exists a non-expansive linear operator π : G → G with ker π ≃ G and 1 For every separable Banach space X and � T � ≤ 1, ∃ i π G G T i X 2 ∀ ε > 0, E ⊆ F finite dimensional Banach spaces, � T � ≤ 1, π G G e T E F Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 10 / 17

An important tool: A universal operator on G There exists a non-expansive linear operator π : G → G with ker π ≃ G and 1 For every separable Banach space X and � T � ≤ 1, ∃ i π G G T i X 2 ∀ ε > 0, E ⊆ F finite dimensional Banach spaces, � T � ≤ 1, ∃ f π G G f e T E F Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 10 / 17