Some notions of compactness in Functional Analysis and one related question about diametral dimensions

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Some notions of compactness in Functional

Analysis and one related question about diametral

dimensions

Loïc Demeulenaere (FRIA-FNRS Grantee)

Comprehensible Seminars - ULiège

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Introduction Some notions of compactness Diametral dimensions

Introduction

Some notions of compactness

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Introduction

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Mathematical Analysis

Main topics: functions and related notions, e.g. limits, distributions, measures, etc.

After algebraic equations, functional equations: dierentiable equations, integral equations, optimization problems...

Study of sets of functions Notions of convergence

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Mathematical Analysis

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Mathematical Analysis

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Mathematical Analysis

Study of sets of functions

Notions of convergence

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Mathematical Analysis

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Mathematical Analysis

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Examples

If K is a compact subset of Rn_{, if f is a function dened on K}

and if (fm)m∈N0 is a sequence of C0(K ) s.t.

sup

x ∈K

|f_m(x ) − f (x )| →0 if m → ∞, then f ∈ C0(K ).

; Convergence in C0(K ) endowed with the metric dened by supK|.|.

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Examples

If K is a compact subset of Rn_{, if f is a function dened on K}

and if (fm)m∈N0 is a sequence of C0(K ) s.t.

sup

x ∈K

|f_m(x ) − f (x )| →0 if m → ∞, then f ∈ C0(K ).

; Convergence in C0(K ) endowed with the metric dened by supK|.|.

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Examples

Lebesgue's dominated convergence theorem: if (fm)m∈N0

is a sequence of L1(R) which converges pointwise to f and if there exists F ∈ L1_{(R) with |f}

m| ≤ F a.e. on R ∀m, then

f ∈L1(R) and Z

R

|fm(x ) − f (x )| dx →0 if m → ∞.

; Convergence in L1(R) for the metric dened by R

R|.| dx. Fourier series: convergence in L2([a, b]) for the metric

dened by qRb

a |.|2dx. ...

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Examples

f ∈L1(R) and Z

R

|fm(x ) − f (x )| dx →0 if m → ∞.

R|.| dx.

Fourier series: convergence in L2([a, b]) for the metric dened by qRb

a |.|2dx. ...

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Examples

f ∈L1(R) and Z

R

|fm(x ) − f (x )| dx →0 if m → ∞.

R|.| dx. Fourier series: convergence in L2_{([a, b])} _{for the metric}

dened by qRb

a |.|2dx. ...

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Examples

f ∈L1(R) and Z

R

|fm(x ) − f (x )| dx →0 if m → ∞.

R|.| dx. Fourier series: convergence in L2_{([a, b])} _{for the metric}

dened by qRb

a |.|2dx. ...

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(Semi)norms

Denition

If E is a vector space on C, a map p : E → [0, ∞) is a seminorm if

1. p(x + y ) ≤ p(x ) + p(y ) ∀x , y ∈ E;

2. p(λx ) = |λ|p(x ) ∀x ∈ E, ∀λ ∈ C.

If p(x) = 0 ⇔ x = 0, then p is a norm.

Property

The (semi)norm p denes a vector (pseudo)metric on E: d (x , y ) = p(x − y ).

; Notions of convergent sequences, Cauchy sequences, etc. Examples

C₀(K ), L1(R), L2([a, b])are (complete) normed spaces (i.e. Banach spaces).

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(Semi)norms

Denition

1. p(x + y ) ≤ p(x ) + p(y ) ∀x , y ∈ E;

2. p(λx ) = |λ|p(x ) ∀x ∈ E, ∀λ ∈ C. If p(x) = 0 ⇔ x = 0, then p is a norm.

Property

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(Semi)norms

Denition

1. p(x + y ) ≤ p(x ) + p(y ) ∀x , y ∈ E;

Property

; Notions of convergent sequences, Cauchy sequences, etc.

Examples

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(Semi)norms

Denition

1. p(x + y ) ≤ p(x ) + p(y ) ∀x , y ∈ E;

Property

C₀(K ), L1(R), L2([a, b]) are (complete) normed spaces (i.e. Banach spaces).

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In general

In general, one single (semi)norm is not sucient...

If Ω is an open subset of Rn _{and if (f}

m)m∈N0 is a sequence of

C₀(Ω)which uniformly converges to f on every compact of Ω, then f ∈ C0(Ω).

; Convergence dened by a family of seminorms supK|.|(K

compact of Ω).

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In general

compact of Ω).

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In general

compact of Ω).

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Locally convex spaces

Denition

A topological vector space (t.v.s.) E is a locally convex space (l.c.s.) if its topology can be dened by a family of seminorms P:

a subset U of E is a neighbourhood (ngbh) of x ∈ E if ∃p ∈ P, ε >0 s.t. {y ∈ E : p(x − y ) ≤ ε} ⊆ U. Remark Good denition if ∀p, q ∈ P, ∃r ∈ P, C > 0 s.t. sup(p(x), q(x)) ≤ Cr(x) ∀x ∈ E .

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Locally convex spaces

Denition

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Locally convex spaces

Denition

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Locally convex spaces

Denition

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Examples of topological properties

For a l.c.s. (E, P) ,

E is Hausdor i ∩p∈Pker(p) = {0};

every x ∈ E has a countable basis of nghbs i P can be chosen countable;

E is metrizable i the two previous points are veried.

Denition

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Examples of topological properties

For a l.c.s. (E, P) ,

E is Hausdor i ∩p∈Pker(p) = {0};

every x ∈ E has a countable basis of nghbs i P can be chosen countable;

E is metrizable i the two previous points are veried.

Denition

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Introduction

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Some notions linked to compactness

Let E be an l.c.s.

A bounded set of E is a subset B of E s.t., for every 0-ngbh U, ∃λ > 0 : B ⊆ λU.

If U, V is 2 subsets of E, V is precompact with respect to U if, ∀ε > 0, ∃P nite subset of E s.t.

V ⊆ εU + P.

K ⊆ E is precompact if it is precompact with respect to each 0-ngbh.

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Some notions linked to compactness

Let E be an l.c.s.

V ⊆ εU + P.

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Some notions linked to compactness

Let E be an l.c.s.

V ⊆ εU + P.

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Links

Proposition

Compact =⇒ Precompact =⇒ Bounded.

Remarks

A precompact set is not always compact, but, in complete spaces, a set is precompact i its closure is compact.

The closed unit ball of an innite-dimensional normed space is never precompact. In particular, a closed bounded set is not always (pre)compact.

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Links

Proposition

Remarks

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Links

Proposition

Remarks

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Two important classes of Fréchet spaces

Let E be a Fréchet space.

E is Montel if every bounded set is precompact.

E is Schwartz if, for every 0-ngbh U, there exists a 0-ngbh V which is precompact with respect to U.

Examples

If Ω is an open set of C, H(Ω) is Schwartz and Montel.

An innite-dimensional Banach space is neither Montel, nor Schwartz.

Warning!

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Two important classes of Fréchet spaces

Examples

Warning!

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Two important classes of Fréchet spaces

Examples

Warning!

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Two important classes of Fréchet spaces

Examples

Warning!

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Two important classes of Fréchet spaces

Examples

Warning!

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Introduction

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Isomorphisms and topological invariants

Sometimes, we have to compare the structures of l.c.s./t.v.s.

Denition

An isomorphism of t.v.s. is a linear, bijective, continuous, and open map between 2 t.v.s.

Question: when are 2 t.v.s. non-isomorphic?

; Topological (linear) invariants: properties preserved by isomorphisms

Examples

Dimension in linear algebra, being Hausdor in topological spaces, etc.

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Isomorphisms and topological invariants

Denition

Examples

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Isomorphisms and topological invariants

Denition

Examples

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Isomorphisms and topological invariants

Denition

Examples

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Isomorphisms and topological invariants

Denition

Examples

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Isomorphisms and topological invariants

Denition

Examples

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Kolmogorov's diameters

Let E be a vector space, V , U ⊆ E, V ⊆ λU (λ > 0).

Denition

The nth _{Kolmogorov's diameters of V with respect to U is}

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

Important property

If U is a ball centred at 0 and associated to a seminorm, then V is precompact with respect to U ⇔ δn(V , U) →0.

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Kolmogorov's diameters

Denition

Important property

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Kolmogorov's diameters

Denition

Important property

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Diametral dimension

Let E be a t.v.s. and U be a basis of 0-nghbs.

Denition

The diametral dimension of E is ∆(E ) =

n

ξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ}_n_δ_n_{(V , U) →}₀o_.

Properties

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

2. If E is Fréchet,

if E is not Schwartz, ∆(E) = c0;

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Diametral dimension

Denition

n

Properties

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Diametral dimension

Denition

n

Properties

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Another diametral dimension...

Denition ∆b(E ) = n ξ ∈ CN0 _{: ∀U ∈ U , ∀B} _{bounded, ξ}_n_δ_n_{(B, U) →}₀o_. Open question Do we have ∆(E ) = ∆b(E ) if E is Fréchet?

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Another diametral dimension...

Denition ∆b(E ) = n ξ ∈ CN0 _{: ∀U ∈ U , ∀B} _{bounded, ξ}_n_δ_n_{(B, U) →}₀o_. Open question Do we have ∆(E ) = ∆b(E ) if E is Fréchet?

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Introduction Some notions of compactness Diametral dimensions Proposition If E is Fréchet, if E is not Montel, ∆b(E ) = c0; if E is Montel, l∞⊆ ∆b(E ). Consequences

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

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

Do we have

∆(E ) = ∆b(E )

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If E is Montel but not Schwartz, then ∆(E) = c0 ( ∆b(E ).

New open question

Do we have

∆(E ) = ∆b(E )

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

Do we have

∆(E ) = ∆b(E )

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A positive partial result

Slight variations of diametral dimensions...

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

ξ ∈ CN0 _{: ∀U ∈ U , ∀B} _{bounded, (ξ}_n_δ_n_{(B, U))}

n∈ `∞

o .

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

If E is Fréchet-Schwartz, then

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

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A positive partial result

n∈ `∞

o .

∆∞(E ) = ∆∞_b (E ).

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

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A positive partial result

n∈ `∞

o .

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

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Schwartz spaces with ∆(E) = ∆

∞

_{(E )}

Classic sequence spaces (Köthe echelon spaces) (2017, F. Bastin, L.D.);

Hilbertizable spaces (in particular nuclear spaces)(2016, L.D., L. Frerick, J. Wengenroth).

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Schwartz spaces with ∆(E) = ∆

∞

_{(E )}

Classic sequence spaces (Köthe echelon spaces)(2017, F. Bastin, L.D.);

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Schwartz spaces with ∆(E) = ∆

∞

_{(E )}

Classic sequence spaces (Köthe echelon spaces)(2017, F. Bastin, L.D.);

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Another sucient condition

Denition (2013, T. Terzio§lu)

A bounded set B of a Fréchet space E is prominent if, for every 0-ngbh U, there exist a 0-nghb V and C > 0 s.t. ∀n

δn(V , U) ≤ C δn(B, V ).

Proposition (2013, T. Terzio§lu)

If E has a prominent bounded set, then ∆(E) = ∆b(E ), but the

converse is false.

Spaces with prominent bounded sets (2016, L.D., L.F., J.W.)

Fréchet spaces with Property Ω: if U = (Uk)k,

∀m, ∃k, ∀j, ∃C >0, ∀r > 0, U_k ⊆ rU_m+C rUj (the converse is false).

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Another sucient condition

δn(V , U) ≤ C δn(B, V ). Proposition (2013, T. Terzio§lu)

converse is false.

∀m, ∃k, ∀j, ∃C >0, ∀r > 0, U_k ⊆ rU_m+C rUj (the converse is false).

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Another sucient condition

δn(V , U) ≤ C δn(B, V ). Proposition (2013, T. Terzio§lu)

converse is false.

∀m, ∃k, ∀j, ∃C >0, ∀r > 0, Uk ⊆ rUm+

C rUj (the converse is false).

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And for non-Fréchet/non-metrizable spaces?

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

There exists a family of Schwartz (and/or nuclear), non-metrizable, l.c.s. E with

∆(E ) 6= ∆b(E ).

Main idea: considering spaces for which the linear span of each bounded set is nite-dimensional.

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And for non-Fréchet/non-metrizable spaces?

∆(E ) 6= ∆b(E ).

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And for non-Fréchet/non-metrizable spaces?

∆(E ) 6= ∆b(E ).

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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):95107, 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):271280, 2016.

H. Jarchow.

Locally Convex Spaces.

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References II

R. G. Meise and D. Vogt.

Introduction to Functional Analysis.

Number 2 in Oxford Graduate Texts in Mathematics. Clarendon Press, Oxford, 1997.

Translated from German by M.S. Ramanujan.

T. Terzio§lu.

Quasinormability and diametral dimension. Turkish J. Math., 37(5):847851, 2013.

Some notions of compactness in Functional Analysis and one related question about diametral dimensions