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 = xE 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 = xE for all x ∈ E is called an isometric embedding. Given ε > 0, a linear mapping f : E → F, with (1 + ε)−1xE ≤ f (x)F ≤ (1 + ε)xE 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 G

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 G

e f

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 G

e f

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 G

e f

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 = xE,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 = xE,i for all i ∈ N and all x ∈ E is called an isometric embedding. Given ε > 0, a linear mapping f : E → F, with (1 + ε)−1xE,i ≤ f (x)F,i ≤ (1 + ε)xE,i 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 GN. Can we find a suitable sequence of semi-norms on GN?

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 : xi = 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 : xi = 0} is a subspace. · i is a norm on Ei := 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 : xi = 0} is a subspace. · i is a norm on Ei := E/ ker · i. (Ei, · i) is a Banach space and E ֒ →

• i

Ei and Ei+1 ։ Ei

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 : xi = 0} is a subspace. · i is a norm on Ei := E/ ker · i. (Ei, · i) is a Banach space and E ֒ →

• i

Ei and Ei+1 ։ Ei If f : E → F is (ε-)isometric, so is fi : Ei → Fi, defined by E F Ei Fi

f can can fi

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,

G G X

π T

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

G G X

π T i

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

G G X

π T i 2 ∀ε > 0, E ⊆ F finite dimensional Banach spaces, T ≤ 1,

G G E F

π e T

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

G G X

π T i 2 ∀ε > 0, E ⊆ F finite dimensional Banach spaces, T ≤ 1, ∃f

G G E F

π e T 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

G G X

π T i 2 ∀ε > 0, E ⊆ F finite dimensional Banach spaces, T ≤ 1, ∃f

G G E F

π e T f

(1) implies that π is a projection (take T = IdG)

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

G G X

π T i 2 ∀ε > 0, E ⊆ F finite dimensional Banach spaces, T ≤ 1, ∃f

G G E F

π e T f

(1) implies that π is a projection (take T = IdG) Based on a result by Cabello Sánchez, Garbulińska-Węgrzyn, Kubiś (2014).

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 10 / 17

A graded sequence of semi-norms

We construct inductively a sequence of semi-norms on GN. For x = (x1, x2, . . .) ∈ GN we define x1 := x1G.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 11 / 17

A graded sequence of semi-norms

We construct inductively a sequence of semi-norms on GN. For x = (x1, x2, . . .) ∈ GN we define x1 := x1G. From the properties of π, we may conclude that G = (im π) ⊕ (ker π) ≃ G ⊕ G = G × G holds isometrically and that there is a norm · ′

2 on G2 such that

G2 is isometric to G and x1 = x1G = π(x1, x2)G ≤ (x1, x2)′

2.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 11 / 17

A graded sequence of semi-norms

We construct inductively a sequence of semi-norms on GN. For x = (x1, x2, . . .) ∈ GN we define x1 := x1G. From the properties of π, we may conclude that G = (im π) ⊕ (ker π) ≃ G ⊕ G = G × G holds isometrically and that there is a norm · ′

2 on G2 such that

G2 is isometric to G and x1 = x1G = π(x1, x2)G ≤ (x1, x2)′

2.

Define x2 := (x1, x2)′

2 and inductively · n in a similar way.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 11 / 17

GN is of almost universal disposition Proposition

The space (GN, { · i}i∈N) is a graded Fréchet space of almost universal disposition for finite dimensional graded Fréchet spaces, i.e., for all ε > 0 and for all finite dimensional graded Fréchet spaces E ⊆ F and all isometric embeddings f : E → GN there is an ε-isometric embedding g : F → GN such that g ↾ E = f .

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 12 / 17

Proof sketch

Given ε > 0, finite dimensional graded Fréchet spaces E and F and an isomeric embedding f : E → F, choose a sequence (εi)i∈N with εi > 0 and

∞

• i=1

(1 + εi) < 1 + ε. Set Ei = (E/ ker · i, · i) and Fi = (F/ ker · i, · i) and fi : Ei → Gi defined by E G Ei Gi.

f can can fi

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 13 / 17

Proof sketch II: Defining g inductively

For i = 1, we have E1 G F1

f1 π

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 14 / 17

Proof sketch II: Defining g inductively

For i = 1, we have E1 G F1

f1 g1 π

g1 is an ε1-isometry.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 14 / 17

Proof sketch II: Defining g inductively

For i = 2, we have E1 G F1 E2 G × G F2

f1 g1 f2 π

g1 is an ε1-isometry.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 14 / 17

Proof sketch II: Defining g inductively

For i = 2, we have E1 G F1 E2 G × G F2

f1 g1 f2 π

g1 is an ε1-isometry.

2 ∀ε > 0, E ⊆ F finite dimensional Banach spaces, T ≤ 1

G G E F

π e T f

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 14 / 17

Proof sketch II: Defining g inductively

For i = 2, we have E1 G F1 E2 G × G F2

f1 g1 f2 π T

g1 is an ε1-isometry.

2 ∀ε > 0, E ⊆ F finite dimensional Banach spaces, T ≤ 1

G G E F

π e T f

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 14 / 17

Proof sketch II: Defining g inductively

For i = 2, we have E1 G F1 E2 G × G F2

f1 g1 f2 π

g1 is an ε1-isometry.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 14 / 17

Proof sketch II: Defining g inductively

For i = 2, we have E1 G F1 E2 G × G F2

f1 g1 f2 π

g1 is an ε1-isometry. Since

• g1 ◦ p2

1

• > 1 is possible, we need an

additional step.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 14 / 17

Proof sketch III: The additional step Lemma

Let X ⊂ Y and A be finite dim. Banach spaces, Z a Banach space, e : X ֒ → A, T : Y → Z with T < r, r > 1, and π: A → Z, π ≤ 1, s.t. A Z X Y

π e T

There exists a finite dim. Banach space C, iA : A ֒ → C, an (r − 1)-isometric embedding iY : Y → C and π′ : C → Z, π′ ≤ 1 s.t. we get the commutative diagram Z A C X Y

π iA π′ e T iY

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π T

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π T

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π T

G × G G E2 F2

π f2 T

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π T

G × G G f2(E2) E2 F2

π π↾f2(E2) f2 T

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π T

G × G G f2(E2) C E2 F2

π π↾f2(E2) T ′ f2 T

T ′ ≤ 1

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π T

G × G G f2(E2) C E2 F2

π π↾f2(E2) T ′ f2 T

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π T g2

G × G G f2(E2) C E2 F2

π π↾f2(E2) T ′ f2 T g2

The map g2 satisfies (1 + ε2)−1(1 + ε1)−1x2 ≤ g2(x)2 ≤ (1 + ε2)(1 + ε1)x2.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π g2

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Proof sketch III: The additional step

E1 G F1 E2 G × G F2

f1 g1 f2 π g2

Then we continue inductively.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 15 / 17

Properties of GN / Comparison to the Banach space case

The space GN is unique up to isometries.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 16 / 17

Properties of GN / Comparison to the Banach space case

The space GN is unique up to isometries. It is universal for separable (graded) Fréchet spaces.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 16 / 17

Properties of GN / Comparison to the Banach space case

The space GN is unique up to isometries. It is universal for separable (graded) Fréchet spaces. There is no separable Fréchet space of universal disposition (where the extension of isometric embeddings would be an isometric embedding and not just ε-isometric).

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 16 / 17

Properties of GN / Comparison to the Banach space case

The space GN is unique up to isometries. It is universal for separable (graded) Fréchet spaces. There is no separable Fréchet space of universal disposition (where the extension of isometric embeddings would be an isometric embedding and not just ε-isometric). No space of the form C(X), where X is a hemi-compact topological space, is of almost universal disposition for finite dimensional (graded) Fréchet spaces.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 16 / 17

References

Christian Bargetz, Jerzy Kąkol, and Wiesław Kubiś. A separable Fréchet space of almost universal dispostition. Preprint, 2016. Félix Cabello Sánchez, Joanna Garbulińska-We ¸grzyn, and Wiesław Kubiś. Quasi-Banach spaces of almost universal disposition.

• J. Funct. Anal., 267(3):744–771, 2014.
• V. I. Gurari˘

ı. Spaces of universal placement, isotropic spaces and a problem

• f Mazur on rotations of Banach spaces.
• Sibirsk. Mat. Ž., 7:1002–1013, 1966.

Wiesław Kubiś and Sławomir Solecki. A proof of uniqueness of the Gurari˘ ı space. Israel J. Math., 195(1):449–456, 2013.

Christian Bargetz (University of Innsbruck) A universal separable Fréchet space 17 / 17