Rough convergence and Chebyshev centers in Banach spaces

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Rough convergence and Chebyshev centers in Banach spaces

Mar´ıa del Carmen List ´an Garc´ıa

9 septiembre 2014 Antequera VI CIDAMA

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I NTRODUCTION AND BACKGROUND

The concept of rough convergence was introduced by H. X. Phu, in finite-dimensional normed spaces and subsequently studied in the infinite-dimensional setting.

The basic idea consists of providing a sort of “generalized limit” for bounded sequences.

Namely, consider a “roughness index” r ≥ 0 and a bounded sequence (xn)_nin a normed space X . Then (xn)_nis said to r -converge to y ∈ X if

lim sup

n

kx_n− y k ≤ r .

Accordingly, the following notation is adopted: lim^rx_n= {y ∈ X : lim sup

n

kx_n− y k ≤ r }

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I NTRODUCTION AND BACKGROUND

lim sup

n

kx_n− y k ≤ r .

n

kx_n− y k ≤ r }

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I NTRODUCTION AND BACKGROUND

lim sup

n

kx_n− y k ≤ r .

n

kx_n− y k ≤ r }

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I NTRODUCTION AND BACKGROUND

lim sup

n

kx_n− y k ≤ r . Accordingly, the following notation is adopted:

lim^rx_n= {y ∈ X : lim sup

n

kx_n− y k ≤ r }

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I NTRODUCTION AND BACKGROUND

lim sup

n

kx_n− y k ≤ r . Accordingly, the following notation is adopted:

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I NTRODUCTION AND BACKGROUND

and of particular interest is the infimum of the roughness indices such that the rough set is not empty, in other words,

¯r (x_n) = ¯r = inf{r > 0 : lim^rx_n6= ∅}.

or equivalently

¯r = inf

y ∈Xlim sup

n

kx_n− y k.

Phu proved that the set lim^rxnis bounded, closed and convex for every r . Moreover, he also studied the set lim^¯^rxnand proved that it has empty interior in finite-dimensional spaces and is nonempty in reflexive spaces.

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I NTRODUCTION AND BACKGROUND

¯r (x_n) = ¯r = inf{r > 0 : lim^rx_n6= ∅}.

or equivalently

¯r = inf

y ∈Xlim sup

n

kx_n− y k.

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I NTRODUCTION AND BACKGROUND

¯r (x_n) = ¯r = inf{r > 0 : lim^rx_n6= ∅}.

or equivalently

¯r = inf

y ∈Xlim sup

n

kx_n− y k.

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I NTRODUCTION AND BACKGROUND

¯r (x_n) = ¯r = inf{r > 0 : lim^rx_n6= ∅}.

or equivalently

¯r = inf

y ∈Xlim sup

n

kx_n− y k.

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I NTRODUCTION AND BACKGROUND

Our goals with this concept were:

Define two new properties based on this concept and study which classical Banach spaces have it.

Show that the behaviour of this set is closely related to Chebyshev centers.

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I NTRODUCTION AND BACKGROUND

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I NTRODUCTION AND BACKGROUND

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D EFINITIONS AND OUR BIG MISTAKE

Two new properties (at least, we thought so)

DEFINITIONS

Let X be a Banach space. We will say that X has

theempty rough interior (ERI) property if int(lim^¯^rx_n) = ∅ for every bounded sequence (x_n)_n⊆ X .

thenonempty rough limit (NRL) property if lim^¯^rx_n6= ∅ for every bounded sequence (x_n)_n⊆ X .

The concepts of lim^¯^rx_n, ¯r and the NRL property were well-known and they are due to M. Edelstein, as follows:

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D EFINITIONS AND OUR BIG MISTAKE

Two new properties (at least, we thought so)

DEFINITIONS

Let X be a Banach space. We will say that X has

theempty rough interior (ERI) property if int(lim^¯^rx_n) = ∅ for every bounded sequence (x_n)_n⊆ X .

thenonempty rough limit (NRL) property if lim^¯^rx_n6= ∅ for every bounded sequence (x_n)_n⊆ X .

The concepts of lim^¯^rx , ¯r and the NRL property were well-known and

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DEFINITIONS

Let X be a normed space. If (xn)_nis a bounded sequence in X , its asymptotic radius is given by

ar (xn) = inf

y ∈Xlim sup

n

kx_n− y k

= ¯r (xn)

and itsasymptotic center by ac(x_n) = {y ∈ X : lim sup

n

kx_n− y k = ar (x_n)}= lim^¯^rx_n.

And we will say that a Banach space issequentially asymptotically center-complete (sacc) whenever every bounded sequence has nonempty asymptotic center= NRL property.

And, inspired by the classical notation, we will say that a Banach space issequentially asymptotically boundary-center (sabc) whenever every bounded sequence has a boundary asymptotic center= ERI property.

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DEFINITIONS

ar (xn) = inf

y ∈Xlim sup

n

kx_n− y k= ¯r (xn)

and itsasymptotic center by ac(x_n) = {y ∈ X : lim sup

n

kx_n− y k = ar (x_n)}= lim^¯^rx_n.

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DEFINITIONS

ar (xn) = inf

y ∈Xlim sup

n

and itsasymptotic center by ac(xn) = {y ∈ X : lim sup

n

kx_n− y k = ar (x_n)}

= lim^¯^rx_n.

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DEFINITIONS

ar (xn) = inf

y ∈Xlim sup

n

kx_n− y k = ar (x_n)}= lim^¯^rxn.

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DEFINITIONS

ar (xn) = inf

y ∈Xlim sup

n

And we will say that a Banach space issequentially asymptotically center-complete (sacc) whenever every bounded sequence has nonempty asymptotic center

= NRL property.

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DEFINITIONS

ar (xn) = inf

y ∈Xlim sup

n

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DEFINITIONS

ar (xn) = inf

y ∈Xlim sup

n

= ERI property.

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DEFINITIONS

ar (xn) = inf

y ∈Xlim sup

n

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R ESULTS ON THE sabc

Let us also recall some well-known concepts in geometry of Banach spaces.

DEFINITIONS

Let X be a Banach space.

If z ∈ S_X, it is said that X isuniformly rotund in the direction z (UR_z) if for every ε > 0 there exists δ > 0 such that if x , y ∈ S_X, x − y ∈ span({z}) and kx − y k ≥ ε then ¹₂kx + y k < 1 − δ.

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R ESULTS ON THE sabc

DEFINITIONS

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R ESULTS ON THE sabc

DEFINITIONS

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DEFINITIONS

X isuniformly rotund in every direction (URED) if it is UR_z for every z ∈ SX.

X hasproperty M if for every u, v ∈ X with kuk = kv k we have lim sup

n

ku + x_nk = lim sup

n

kv + x_nk

whenever x_n−→ 0.^w

Let 1 ≤ p ≤ ∞, X is said to haveproperty m_pif lim sup

n

kx + x_nk = k(kxk, lim sup

n

kx_nk)k_p

whenever xn w

−→ 0 and x ∈ X .

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DEFINITIONS

n

ku + x_nk = lim sup

n

kv + x_nk

Let 1 ≤ p ≤ ∞, X is said to haveproperty m_pif lim sup

n

kx_nk)k_p

whenever xn w

−→ 0 and x ∈ X .

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DEFINITIONS

n

ku + x_nk = lim sup

n

kv + x_nk

Let 1 ≤ p ≤ ∞, X is said to haveproperty mpif lim sup

n

kx_nk)k_p

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PROPOSITION

Let X be a Banach space. If X is URz for some z ∈ SX then X is sequentially asymptotically boundary-center.

EXAMPLES

C(K ) is URξ1 for every compact space K , where ξ₁:K → R is the function constantly equal to 1.

`₁is URe1.

`_pspaces for 1 < p < ∞ are known to be uniformly rotund.

PROPOSITION

Every finite-dimensional space, as well as C(K ) and `pare

sequentially asymptoticaly boundary-center whenever K is a compact space and 1 ≤ p ≤ ∞.

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PROPOSITION

EXAMPLES

`₁is URe1.

PROPOSITION

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PROPOSITION

EXAMPLES

`₁is URe1.

PROPOSITION

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PROPOSITION

EXAMPLES

`₁is URe1.

PROPOSITION

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PROPOSITION

EXAMPLES

`₁is URe1.

PROPOSITION

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However, there are also spaces which lack the property.

EXAMPLE

c0is not sabc.

If we consider the canonical sequence (e_n)_nit is straightforward that ar (e_n) =1 and every y ∈ B_c₀ satisfies lim sup_nky − e_nk = 1. Moreover, if y ∈ c₀but ky k > 1 then we will have

lim sup

n

ky − e_nk = ky k > 1.

So ac(e_n) =B_c₀ which has nonempty interior.

Let us recall that if L is a locally compact, noncompact space then ∞ denotes the only point added in its Alexandroff compactification ˆL, which is assumed to contain L, and C0(L) is the space of continuous functions in L which vanish at infinity.

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EXAMPLE

c0is not sabc.

If we consider the canonical sequence (e_n)_nit is straightforward that ar (e_n) =1 and every y ∈ B_c₀ satisfies lim sup_nky − e_nk = 1. Moreover, if y ∈ c₀but ky k > 1 then we will have

lim sup

n

ky − e_nk = ky k > 1.

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EXAMPLE

c0is not sabc.

If we consider the canonical sequence (en)_nit is straightforward that ar (e_n) =1 and every y ∈ B_c₀ satisfies lim sup_nky − e_nk = 1.

Moreover, if y ∈ c₀but ky k > 1 then we will have lim sup

n

ky − e_nk = ky k > 1.

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EXAMPLE

c0is not sabc.

Moreover, if y ∈ c₀but ky k > 1 then we will have

lim sup

n

ky − e_nk = ky k > 1.

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EXAMPLE

c0is not sabc.

n

ky − e_nk = ky k > 1.

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EXAMPLE

c0is not sabc.

n

ky − e_nk = ky k > 1.

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EXAMPLE

c0is not sabc.

n

ky − e_nk = ky k > 1.

Let us recall that if L is a locally compact, noncompact space then ∞ denotes the only point added in its Alexandroff compactification ˆL,

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PROPOSITION

If L is a locally compact, noncompact space such that ∞ has a countable neighbourhood basis in ˆL then there exists (f_n)_n⊆ S_C₀_(L) such that ac(fn) =B_C₀_(L). In particular, C0(L) is not sabc.

THEOREM

Let X be a Banach space with the M property. If X has an isometric copy of c₀then X is not sabc.

THEOREM

If X is a Banach space that has the m_∞property and lacks the Schur property then X is not sabc.

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PROPOSITION

THEOREM

If X is a Banach space that has the m_∞property and lacks the Schur property then X is not sabc.

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PROPOSITION

THEOREM

If X is a Banach space that has the m∞property and lacks the Schur property then X is not sabc.

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T HE sacc ^AND C HEBYCHEV CENTER

Phu proved that reflexive spaces are sacc. The gist of his argument consists in using the weak compactness of bounded closed convex sets in such spaces.

In this section, we will see what happens with spaces that are sacc. First, we need some definitions.

DEFINITIONS

Let X be a Banach space and consider a bounded nonempty set A ⊆ X .

TheChebyshev radius of A is given by ρ(A) = inf

x ∈Xsup

a∈A

kx − ak. z ∈ X is aChebyshev center for A if sup

a∈A

kz − ak = ρ(A), in other words, the Chebyshev radius is attained at z.

X is said to beChebyshev center complete (cc) if every bounded nonempty set has Chebyshev center.

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T HE sacc ^AND C HEBYCHEV CENTER

In this section, we will see what happens with spaces that are sacc.

First, we need some definitions.

DEFINITIONS

x ∈Xsup

a∈A

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T HE sacc ^AND C HEBYCHEV CENTER

DEFINITIONS

x ∈Xsup

a∈A

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T HE sacc ^AND C HEBYCHEV CENTER

DEFINITIONS

x ∈Xsup

a∈A

kx − ak.

z ∈ X is aChebyshev center for A if sup

a∈A

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T HE sacc ^AND C HEBYCHEV CENTER

DEFINITIONS

x ∈Xsup

a∈A

kx − ak.

a∈A

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T HE sacc ^AND C HEBYCHEV CENTER

DEFINITIONS

x ∈Xsup

a∈A

kx − ak.

a∈A

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The following is a generalization of this concept:

DEFINITIONS

Let X be a Banach space and B = (Bα)_α∈Λa decreasing net of bounded nonempty subsets of X .

Theasymptotic radius of the net is given by r (B) = inf

x ∈Xlim sup

α

{kx − y k : y ∈ Bα}.

The set {x ∈ X : lim sup_α{kx − y k : y ∈ Bα} = r (B)} is called asymptotic center of B.

X is said to beasymptotic center complete (acc) if every decreasing net of bounded nonempty sets has nonempty asymptotic center.

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DEFINITIONS

x ∈Xlim sup

α

{kx − y k : y ∈ Bα}.

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DEFINITIONS

x ∈Xlim sup

α

{kx − y k : y ∈ Bα}.

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DEFINITIONS

x ∈Xlim sup

α

{kx − y k : y ∈ Bα}.

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DEFINITIONS

x ∈Xlim sup

α

{kx − y k : y ∈ Bα}.

X is said to beasymptotic center complete (acc) if every decreasing net of bounded nonempty sets has nonempty

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It is clear that acc implies cc, consider a net with only one element.

acc also implies sacc, just take B_k = {x_n : n ≥ k }.

LEMMA

Let (xn)_n⊆ X be a bounded sequence and for each k ∈ N let C_k = {x_n : n ≥ k }. We have ar (x_n) =lim_kρ(C_k) =inf_kρ(C_k).

LEMMA

If there exists a countable bounded subset in X without Chebyshev center then X is not sacc.

PROPOSITION

Let X be a separable Banach space which is also sacc, then X is cc.

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LEMMA

Let (xn)_n⊆ X be a bounded sequence and for each k ∈ N let C_k = {x_n : n ≥ k }. We have ar (x_n) =lim_kρ(C_k) =inf_kρ(C_k).

LEMMA

PROPOSITION

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acc ⇒ sacc

⇓ cc

LEMMA

Let (x_n)_n⊆ X be a bounded sequence and for each k ∈ N let C_k = {x_n : n ≥ k }. We have ar (x_n) =lim_kρ(C_k) =inf_kρ(C_k). LEMMA

PROPOSITION

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acc ⇒ sacc

⇓6⇑

cc

LEMMA

Let (x_n)_n⊆ X be a bounded sequence and for each k ∈ N let C_k = {x_n : n ≥ k }. We have ar (x_n) =lim_kρ(C_k) =inf_kρ(C_k). LEMMA

PROPOSITION

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acc ⇒ sacc

⇓6⇑

cc

LEMMA

Let (x_n)_n⊆ X be a bounded sequence and for each k ∈ N let C_k = {x_n : n ≥ k }. We have

ar (x_n) =lim_kρ(C_k) =inf_kρ(C_k). LEMMA

PROPOSITION

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acc ⇒ sacc

⇓6⇑

cc

LEMMA

Let (x_n)_n⊆ X be a bounded sequence and for each k ∈ N let C_k = {x_n : n ≥ k }. We have ar (x_n) =lim_kρ(C_k) =inf_kρ(C_k).

LEMMA

PROPOSITION

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acc ⇒ sacc

⇓6⇑

cc

LEMMA

PROPOSITION

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acc ⇒ sacc

⇓6⇑

cc

LEMMA

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THEOREM

Let X be a separable Banach space. If X is sacc then X is acc.

acc + separable

⇐⇒ sacc

⇓6⇑ cc

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THEOREM

Let X be a separable Banach space. If X is sacc then X is acc.

acc + separable

⇐⇒ sacc

⇓6⇑

cc

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R EFERENCIAS

ANDERSON, C.L., HYAMS, W.H.ANDMCKNIGHT, C.K., “Center points of nets”, Can.J. Math.,2(1975), 418–422.

PHU, H. X., “Rough convergence in normed linear spaces”, Numer. Funct. Anal. Optim.22 (2001), 201–224.

PHU, H. X., “Rough convergence in infinite-dimensional normed spaces”, Numer. Funct. Anal. Optim.24 (2003), 284–301.

GARKAVI, A. L., “On the optimal net and best cross-section of a set in a normed space” (Russian), Izv. Akad. Nauk SSSR Ser.

Mat.26 (1962), 87–106.

KALTON, N. J., “M-ideal of compact operators”, Illinois J. Math.

37 (1993), no.1, 147–169.

Rough convergence and Chebyshev centers in Banach spaces