A NOTE ON ( gDF )-SPACES

Certain locally convex spaces of scalar-valued mappings are shown to be finitedimensional.


Introduction. Radenovic
, generalizing a result of Iyahen [2], has shown that if E is a Banach space and (E, σ (E, E )) (or (E ,σ (E ,E))) is a (DF )-space [1], then E is finite-dimensional.His result has been extended to arbitrary locally convex spaces by Krassowska and Sliwa [3].
In [4,5], (DF )-spaces have been generalized as follows: a locally convex space (E, τ) is a (gDF )-space if (a) (E, τ) has a fundamental sequence (B n ) n∈N of bounded sets, and (b) τ is the finest locally convex topology on E that agrees with τ on each B n .In this note, we prove that if an arbitrary vector space of scalar-valued mappings is a (gDF )-space under the locally convex topology of pointwise convergence, then it is finite-dimensional.As a consequence, the above-mentioned theorem of Krassowska and Sliwa readily follows.
2. The result.Throughout this note, all vector spaces under consideration are vector spaces over a field K which is either R or C. In our result, E denotes an arbitrary set and H denotes a subspace of the vector space of all mappings from E into K.We consider on H the separated locally convex topology of pointwise convergence and represent by H the topological dual of H. Theorem 2.1.The following conditions are equivalent: (a) H is a finite-dimensional vector space; Proof.It is clear that (a) implies (b) and (b) implies (c) (every (DF )-space is a (gDF )space).
Suppose that condition (c) holds.If H is infinite-dimensional, there exists a countable linearly independent subset {ϕ n ; n ∈ N} of H . Let (B n ) n∈N be an increasing fundamental sequence of bounded subsets of H.Then, (B 0 n ) n∈N is a decreasing sequence of neighborhoods of zero in (H , β(H ,H)) forming a fundamental system of neighborhoods of zero in (H , β(H ,H)).For each n ∈ N, fix an α n > 0 such that α n ϕ n ∈ B 0 n ; then (α n ϕ n ) n∈N converges to zero in (H , β(H ,H)).By [5, Theorem 1.1.7],the set Γ = {α n ϕ n ; n ∈ N} is equicontinuous.Hence, there exist x 1 ,...,x m ∈ E and there exists an α > 0 such that the relations For each i = 1,...,m, let δ i ∈ H be given by δ where [F ] is the finite-dimensional vector space generated by F .Indeed, let ϕ ∈ Γ and take an f Then, for all λ ∈ K, Consequently, |ϕ(λf Therefore the vector space generated by the set {ϕ n ; n ∈ N} is finite-dimensional, which contradicts the choice of (ϕ n ) n∈N .This completes the proof of the theorem.
Remark 2.2.The theorem of Krassowska and Sliwa mentioned at the beginning of this note follows from Theorem 2.1.In fact, let E be a separated locally convex space.If (E ,σ (E ,E)) is a (DF )-space, then E is finite-dimensional by Theorem 2.1, and so E is finite-dimensional.Hence, E is finite-dimensional if (E, σ (E, E )) is a (DF )-space.

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.