WEBBED SPACES , DOUBLE SEQUENCES , AND THE MACKEY CONVERGENCE CONDITION

In [3], Gilsdorf proved, for locally convex spaces, that every sequentially webbed space satisfies the Mackey convergence condition. In the more general frame of topological vector spaces, this theorem and its inverse are studied. The techniques used are double sequences and the localization theorem for webbed spaces.


Introduction.
A web W in a topological vector space E is a countable family of balanced subsets of E, arranged in layers.The first layer of the web consists of a sequence (A p : p = 1, 2,...) whose union absorbs each point of E. For each set A p of the first layer, there is a sequence (A pq : q = 1, 2,...) of sets, called the sequence determined by A p , such that A pq + A pq ⊂ A p for each q; (1) A pq : q = 1, 2,... absorbs each point of A p . ( Further, layers are made up in a corresponding way such that each set of the kth layer is indexed by a finite row of k integers and, at each step, the above mentioned two conditions are satisfied.Suppose that one chooses a set A p from the first layer, then a set A pq of the sequence determined by A p and so on.The resulting sequence S = (A p ,A pq ,A pqr ,...) is called a strand.Whenever we are dealing with only one strand, we can simplify the notation by writing W 1 = A p , W 2 = A pq , etc. Thus, S = (W k ) is a strand, where, for each k, W k is a set of the kth layer.Let S = (W k ) be a strand.Consider x k ∈ W k and the series x k converges to some x ∈ W n for every n ∈ N and for any choice of x k ∈ W k .The standard references for webs in a topological vector space are [5,7,8].
Let (E, τ) be a topological vector space.(x n ) n ⊂ E is a Mackey null sequence if there exists a sequence of real numbers (r n ) n such that r n → ∞ and r n x n → 0 in E. We say that (x n ) n ⊂ E is Mackey convergent to x if (x n − x) n is a Mackey null sequence.A topological vector space E satisfies the Mackey convergence condition (M.c.c.) if every null sequence is Mackey null.

Double sequences.
A completing double sequence in a topological vector space (E, τ) is a family (K n j ) n,j∈N of balanced subsets such that (1) K n j ⊂ K n+1 j for every n, j natural numbers; (2) K n j+1 + K n j+1 ⊂ K n j for every n, j natural numbers; (3) n∈N K n j is absorbent in E for every j natural number; (4) for every j 0 ∈ N, if x j ∈ K n j with j > j 0 , then ∞ j=j 0 +1 x j converges in E to some x ∈ K n j 0 .Moreover, (K n j ) j,n∈N is compatible with the topology if, for each zero neighborhood U in E and for every natural number n, there exists a natural number J such that K n j ⊂ U for every j ≥ J.For example, if E is sequentially complete and has a fundamental sequence of closed bounded sets A 1 ⊂ A 2 ⊂ ••• such that, for each bounded set B ⊂ E, there exists n 0 ∈ N such that B ⊂ A n 0 (this is the case if E is the strong dual of a metrizable space).In this case, we define K n j = 2 −j A n and it is easy to verify the properties (1) to (4), above.The reader can find further information concerning double sequences in [6].
A topological vector space (E, τ), with a compatible completing double sequence (K n j ), has a Sequential Double Sequence or the SDS property if, for each x m → 0 in E, there exists n 0 ∈ N such that, for each j, there exists a natural number M j such that Theorem 1.Let (E, τ) be a topological vector space with the SDS property.Then E satisfies the Mackey convergence condition.
Proof.Let x m → 0 in (E, τ).Let (K n j ) be a sequential double sequence, then there exists n 0 ∈ N such that, for every j, there exists a natural number M j such that x m ∈ K n 0 j , for every m ≥ M j .For n, j ∈ N, we have j+1 , for every m ≥ M 2(j+1) ; and so, for all j ∈ N. Define r m = j if M 2j ≤ m < M 2(j+1) , then lim m→∞ r m = lim j→∞ j = ∞.Since (K n j ) is compatible with the topology, we conclude that r m x m → 0.
From the theorem, a space with the SDS property is a space with the Mackey convergence condition.In what follows, we study the conditions under which we have an equivalence of these two properties.First, let us introduce another type of double sequences: a topological vector space (E, τ), with a compatible completing double sequence (K n j ), has a quasi-Sequential Double Sequence or the qSDS property if, for each x n → 0 in E, there exists n 0 such that, for every j, there exists a natural number M j and a positive real number α j such that m > M j implies that x m ∈ α j K n 0 j .If α j = 1, for every j, in a qSDS, then it becomes on SDS.So, the qSDS is more general than the SDS.The next proposition gives the condition for the equivalence.
Proposition 2. Let (E, τ) be a topological vector space with the Mackey convergence condition.Then the SDS and the qSDS are the same.
Proof.Let x m → 0 in a space (E, τ) with qSDS property.By the Mackey convergence condition, there exists a scalar sequence r m → ∞ such that r m x m → 0. Then there exists n 0 such that r m x m ∈ α j K n 0 j , for some α j > 0 whenever m ≥ M j .Hence, Next, we see an example, where the qSDS property holds and the SDS property does not.
Let (E, • ) be a Banach space with a sequence (x m ) m∈N weakly convergent to zero and not norm convergent.Let B be the closed unit ball in E. For each n, j ∈ N, let K n j = 2 −j B. Then (K n j ) is a compatible completing double sequence with respect to the norm topology and, consequently, with respect to any weaker topology τ, especially the weak topology since the map i : (E, • ) → (E, τ) is continuous.Now, (x m ) m∈N is not contained in K n j , since K n j are neighborhoods in the norm topology such that j K n j = {0} and, by [4,Ex. 4] and [4, cor. of Thm.3], (E, σ ) does not have the M.c.c.Nevertheless, (x m ) m∈N is bounded with respect to both the weak and norm topologies.So, for every K n j , there exists α j such that (x m ) m ⊂ α j K n j .We have the following implication: SDS ⇒ qSDS.This implication can be reversed if the space has the M.c.c.Furthermore SDS ⇒ M.c.c.So, we have the following corollary: Corollary 3. Let E be a topological vector space with a compatible completing double sequence.Then E has SDS property if and only if the qSDS property and M.c.c.hold.

Mackey convergence and sequentially webbed spaces. E is sequentially webbed
if it has a compatible web W such that, for every null sequence (x n ) n∈N in E, there exists a finite collection of strands W k .Gilsdorf [3] proved two relations between the M.c.c. and the sequentially webbed spaces in the locally convex case.
Here, we generalize these results.One to topological vector spaces and the other to locally r-convex spaces.In fact, the concept of webbed spaces, introduced here, does not use local convexity.Note that in this case, in each strand, we have 2W k+1 ⊂ W k+1 + W k+1 ⊂ W k so that W k+1 ⊂ 2 −1 W k , and then following the proof of [3,Thm.12], we have: if (E, τ) is a sequentially webbed topological vector space, then E has the M.c.c.
In order to obtain a converse of this result, we need to use a localization theorem [5, Thm.5.6.
we have the usual convexity definition.
For U ⊂ E balanced and absorbent, let q u : E → R + be the Minkowski functional defined by x → inf{ρ > 0 : x ∈ ρU}.q u is an r-seminorm if q u (x + y) r ≤ q u (x) r + q u (y) r .Furthermore, if q −1 u (0) = 0, it is called an r-norm.(E, τ) is locally r-convex if it has a fundamental system of zero neighborhoods formed by r-convex sets.Now, we can use the E B spaces for locally r-convex spaces.(E, τ) locally r-convex space is locally r-Baire if, for every bounded set A ⊂ E, there exists B absolutely rconvex and bounded such that A ⊂ B and the space (E B ,ρ B ) is a Baire space, where E B is the span of B and ρ B is the topology generated by the r-norm q r B .
Theorem 4. Let (E, τ) be a locally r-Baire locally r-convex space and strictly webbed.If E satisfies the Mackey convergence condition, then E is sequentially webbed.
Proof.Let W be a strict web in E; (x n ) n ⊂ E a null sequence, and r n → ∞ a sequence of real numbers such that r n x n → 0 in E. Let A = {r n x n : n ∈ N}, A is bounded, then there exists a bounded absolutely r-convex set B such that (E B ,ρ B ) is a Baire space and A is a bounded set in E B .The identity map i : E B → E is continuous.Hence, by the localization theorem, i has a closed graph and there exists a strand

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: of W such that, for every natural number k, there exists M k such that n ≥ M k implies x n ∈ Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; edleonel@rc.unesp.brAlexander Loskutov, Physics Faculty, Moscow State University, Vorob'evy Gory, Moscow 119992, Russia; loskutov@chaos.phys.msu.ru