Diametral dimension(s) and prominent bounded sets

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Loïc Demeulenaere (FRIA-FNRS Grantee)

Septièmes journées “Besançon-Neuchâtel” d’Analyse Fonctionnelle: Besançon

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Introduction: Diametral dimension Some first results Prominent bounded sets Non-Fréchet spaces

Introduction: Diametral dimension

Some first results

Prominent bounded sets

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Some first results

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Diametral dimension

Let E be a topological vector space (t.v.s.) and U be a basis of 0-neighbourhoods.

Definition

The diametral dimension of E is ∆(E ) := n ξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U, s.t. ξ} nδn(V , U) → 0 o , with δn(V , U) := inf {δ > 0 : ∃L ⊆ E , dim L ≤ n, s.t. V ⊆ δU + L}. NB

If U is absolutely convex and absorbing, then V is precompact with respect to U iff δn(V , U) → 0.

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Diametral dimension

Definition

The diametral dimension of E is ∆(E ) := n ξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U, s.t. ξ} nδn(V , U) → 0 o , with δn(V , U) := inf {δ > 0 : ∃L ⊆ E , dim L ≤ n, s.t. V ⊆ δU + L}.

NB

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Diametral dimension

Definition

The diametral dimension of E is ∆(E ) := n ξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U, s.t. ξ} nδn(V , U) → 0 o , with δn(V , U) := inf {δ > 0 : ∃L ⊆ E , dim L ≤ n, s.t. V ⊆ δU + L}. NB

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Properties of diametral dimension

Proposition

• ∆ is a topological invariant: if E ∼= F , then ∆(E ) = ∆(F ).

• c0⊆ ∆(E ).

• If E is a l.c.s.,

I E is Schwartz ⇔ c0( ∆(E ) ⇔ `∞⊆ ∆(E );

I E is nuclear ⇔ ∃p > 0 : ∀p > 0, (np₎ n∈ ∆(E ).

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Properties of diametral dimension

Proposition

• c0⊆ ∆(E ).

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Properties of diametral dimension

Proposition

• c0⊆ ∆(E ).

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Properties of diametral dimension

Proposition

• c0⊆ ∆(E ).

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Another diametral dimension...

Definition

If E is a t.v.s. and U is basis of 0-neighbourhoods, ∆b(E ) := n ξ ∈ CN0 _{: ∀B bounded, ∀U ∈ U , ξ} nδn(B, U) → 0 o .

Reminder: ∆(E ) =nξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U s.t. ξ}

nδn(V , U) → 0 o . NB ∆(E ) ⊆ ∆b(E ). Question (Mityagin)

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Another diametral dimension...

Definition

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Another diametral dimension...

Definition

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Some first results

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One observation

Let E a Fréchet space.

Reminder: E is Schwartz ⇔ c0 ( ∆(E ) ⇔ `∞⊆ ∆(E ).

Similarly,

E is Montel ⇔ c0( ∆b(E ) ⇔ `∞⊆ ∆b(E ). Consequences

• If E is not Montel: ∆(E ) = ∆b(E ) = c0.

• If E is Montel but not Schwartz: ∆(E ) = c0( ∆b(E ). New open question

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One observation

Similarly,

E is Montel ⇔ c0( ∆b(E ) ⇔ `∞⊆ ∆b(E ).

Consequences

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One observation

Similarly,

• If E is Montel but not Schwartz: ∆(E ) = c0( ∆b(E ).

New open question

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One observation

Similarly,

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One first result

Notations ∆∞(E ) := n ξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , s.t. (ξ} nδn(V , U))_n∈ `∞ o , ∆∞_b (E ) :=nξ ∈ CN0 _{: ∀B bounded, ∀U ∈ U , (ξ} nδn(B, U))n∈ `∞ o .

Reminders: ∆(E ) =n_{ξ ∈ C}N0_{: ∀U ∈ U , ∃V ∈ U s.t. ξ}

nδn(V , U) → 0 o and ∆b(E ) = n ξ ∈ CN0_{: ∀B bounded, ∀U ∈ U , ξ} nδn(B, U) → 0 o .

Theorem(2016, L.D., L. Frerick, J. Wengenroth) If E is a Fréchet-Schwartz, then ∆∞(E ) = ∆∞_b (E ).

In particular, if ∆(E ) = ∆∞_{(E ), then ∆(E ) = ∆} b(E ).

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One first result

In particular, if ∆(E ) = ∆∞_{(E ), then ∆(E ) = ∆} b(E ).

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One first result

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∆(E ) = ∆

∞

(E )

Which Fréchet-Schwartz spaces verify ∆(E ) = ∆∞(E )?

• HilbertizableFréchet-Schwartz spaces and, in particular, nuclearFréchet spaces (2016, L.D., L. Frerick, J. Wengenroth);

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∆(E ) = ∆

∞

(E )

Which Fréchet-Schwartz spaces verify ∆(E ) = ∆∞(E )? • Hilbertizable Fréchet-Schwartz spaces and, in particular,

nuclear Fréchet spaces (2016, L.D., L. Frerick, J. Wengenroth);

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Some first results

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Another idea...

We want ∆b(E ) = n ξ ∈ CN0 _{: ∀B bounded, ∀U ∈ U , ξ} nδn(B, U)→ 0 o ⊆∆(E ) = n ξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U s.t. ξ}_nδ n(V , U)→ 0 o .

Prominent bounded sets (2013, T. Terzioglu)

A bounded B set of E is prominent if ∀U ∈ U , ∃V ∈ U , C > 0 s.t. δn(V , U) ≤ C δn(B, V ) ∀n ∈ N0.

Property

If E has a prominent set, then ∆(E ) = ∆b(E ). Warning!

The converse is false (e.g. for power series spaces of infinite type).

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Another idea...

Property

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Another idea...

Property

If E has a prominent set, then ∆(E ) = ∆b(E ).

Warning!

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Another idea...

Property

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Looking for prominent sets...

Reminder

A Fréchet space E , with a fundamental system of semi-norms (k.kn)n∈N, has the property Ω if

∀m ∃k ∀j ∃C > 0 : kx0k∗_k2

≤ C kx0k∗_mkx0k∗_j ∀x0 ∈ E0 where k.k∗_m is the dual norm of k.km.

Proposition (2017, F. Bastin, L.D. ; 2016, L.D., L. Frerick, J. Wengenroth)

If E has the property Ω, then it has a prominent bounded set. In particular, ∆(E ) = ∆_b(E ).

Remarks (2016, L.D., L. Frerick, J. Wengenroth)

• For regular Köthe spaces, existence of prominent sets ⇔ Ω. • There exist spaces with prominent sets but without Ω (e.g.

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Looking for prominent sets...

Reminder

∀m ∃k ∀j ∃C > 0 : kx0k∗_k2

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Looking for prominent sets...

Reminder

∀m ∃k ∀j ∃C > 0 : kx0k∗_k2

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Some first results

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Counterexamples

Main idea

If E is a l.c.s. s.t. every bounded set is “finite-dimensional”, then ∆b(E ) = CN0. Reminders: ∆b(E ) = n ξ ∈ CN0 _{: ∀B bounded, ∀U ∈ U , ξ} nδn(B, U) → 0 o

and δn(B, U) = inf {δ > 0 : ∃L ⊆ E , dim L ≤ n, s.t. B ⊆ δU + L}.

Purpose

Finding a family of Schwartz spaces E with only finite-dimensional bounded sets s.t. ∆(E ) 6= CN0_...

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Counterexamples

Main idea

If E is a l.c.s. s.t. every bounded set is “finite-dimensional”, then ∆b(E ) = CN0. Reminders: ∆b(E ) = n ξ ∈ CN0 _{: ∀B bounded, ∀U ∈ U , ξ} nδn(B, U) → 0 o

and δn(B, U) = inf {δ > 0 : ∃L ⊆ E , dim L ≤ n, s.t. B ⊆ δU + L}. Purpose

Finding a family of Schwartz spaces E with only finite-dimensional bounded sets s.t. ∆(E ) 6= CN0_...

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Interesting properties

Proposition (1979, A. Wilansky)

A Hausdorff l.c.s. E has only finite-dimensional bounded sets if and only if any linear functional on E is locally bounded.

In particular, if E∗ is the algebraic dual of E , all the bounded sets of σ(E , E∗) are finite-dimensional.

Proposition (1981, H. Jarchow)

If E and F are two l.c.s., F barrelled, for which ∃T : E → F linear, continuous, and surjective, then ∆(E ) ⊆ ∆(F ).

In particular, if T1 and T2 are 2 l.c.t. on a vector space E , with T1 barrelled and T1 ≤ T2, then

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Interesting properties

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Interesting properties

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Interesting properties

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Counterexamples

Theorem(2017, F. Bastin, L.D.)

If E is a vector space and

• T₁ is a barrelled l.c.t. on E s.t. ∆(E , T1) ( CN0 (e.g. E ≡

Köthe spaces),

• T₂ is a l.c.t. with only finite-dimensional bounded sets (e.g. σ(E , E∗)),

• T is the projective limit of T1 and T2,

then

∆(E , T ) ( ∆b(E , T ).

In particular, if T1 and T2 are Schwartz (resp. nuclear), then T is

Schwartz (resp. nuclear).

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Counterexamples

Köthe spaces),

then

∆(E , T ) ( ∆b(E , T ).

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Counterexamples

Köthe spaces),

then

∆(E , T ) ( ∆b(E , T ).

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Counterexamples

Köthe spaces),

then

∆(E , T ) ( ∆b(E , T ).

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Counterexamples

Köthe spaces),

then

∆(E , T ) ( ∆b(E , T ).

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References I

F. Bastin and L. Demeulenaere.

On the equality between two diametral dimensions.

Functiones et Approximatio, Commentarii Mathematici, 56(1):95–107, 2017.

L. Demeulenaere.

Dimension diamétrale, espaces de suites, propriétés (DN) et (Ω).

Master’s thesis, University of Liège, 2014.

L. Demeulenaere, L. Frerick, and J. Wengenroth. Diametral dimensions of Fréchet spaces.

Studia Math., 234(3):271–280, 2016.

H. Jarchow.

Locally Convex Spaces.

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References II

T. Terzioglu.

Diametral Dimension and Köthe Spaces.

Turkish J. Math., 32(2):213–218, 2008.

T. Terzioglu.

Quasinormability and diametral dimension.

Turkish J. Math., 37(5):847–851, 2013.

D. Vogt.

Lectures on Fréchet spaces.

Lecture Notes, Bergische Universität Wuppertal, 2000.

A. Wilansky.

Modern Methods in Topological Vector Spaces.