Visualization of neighbourhoods in some FK spaces

Transcription

1 Visualization of neighbourhoods in some FK spaces Vesna Veličković and Eberhard Malkowsky, Faculty of Science and Mathematics, University of Niš, Serbia Department of Mathematics, Fatih University, Istanbul, Turkey Državni Univerzitet u Novom Pazaru, Vuka Karadžića bb, Novi Pazar, Serbia Abstract. In this extended abstract, we present the graphical representations of some neighbourhoods in certain FK spaces that have recently been studied. These visualizations strongly support the understanding of the topological and geometric structures of the spaces. We emphasize that the graphics in this paper were created by our own software package and its extensions [ 4]. Keywords: Visualization, Topologies, Neighborhoods, FK spaces, Dual spaces PACS: Lt, Pc INTRODUCTION We recall some well known facts and definitions. The set ω of all complex sequences x =(x k ) is a Fréchet space, that is, a complete linear metric space with the metric d ω and the algebraic operations of addition and multiplication by scalars defined by d ω (x,y)= 2 k x k y k for all x,y ω, () + x k y k and x + y =(x k + y k ) and λ x =(λx k) for all x,y ω and all λ C; also, convergence in (ω,d ω) and coordinatewise convergence are equivalent, and d ω is the weakest metric for which this is true ([5, Example 9.3.7]). An FK space (X,d) is a complete subspace of ω with its metric d stronger than d ω on X, that is, in which the coordinates P n : X C are continuous where P n (x)=x n (x =(x k ) X);aBK space is an FK space with its metric given by a norm. If X is a normed space then, as usual, X denotes the set of all continuous linear functionals on X. IfX ω, then the β dual of X is the set X β = { a k x k :convergesforallx X. Example (a) Obviously (ω,d ω ) is an FK space and the sets l, c and c 0 of all bounded, convergent and null sequences are BK spaces with their natural norms defined by x = sup k x k in each case ([5, p. 55]). (b) Let p =(p k ) l be a positive sequence, H = p and M = max{,h. Then, (l(p),d (p) ) is an FK space, where l(p)= { x k p k < and d (p) (x,y)= ( x k y k p k ) /M for all x,y l(p); ifp k = p for all k, then l(p) reduces to the familiar BK space l p with its natural norm defined by x p =( x k p ) /p ([6, Example 3.3 (b), (c)]). SPACES Here, we consider the spaces w p 0 (Λ) and wp (Λ) that were defined in [7], their topological properties, and the first and second dual spaces. Advancements in Mathematical Sciences AIP Conf. Proc. 676, ; doi: 0.063/ AIP Publishing LLC /$

2 Throughout, let p < and q be the conjugate number of p, thatis,q = for p = andq = p/(p ) for < p <. A non decreasing sequence Λ =(λ n ) n=0 is exponentially bounded ([7, Lemma ]) if and only if { There are reals s t such that for some subsequence (λn(ν) ) ν=0 0 < s λ n(ν) (2) λ < forν = 0,,... n(ν+) Also let Λ =(λ n ) n=0 be an exponentially bounded sequence, and (λ n(ν)) ν=0 an associated subsequence with λ n(0) = λ 0. We write K <ν> (ν = 0,,...) for the set of all integers k with n(ν) k n(ν + ), and consider the sets { ( ) w p 0 (Λ) = lim ν λ n(ν+) x k p = 0, { ( k K <ν> ) w (Λ) p = sup λ ν n(ν+) x k p <. k K <ν> The first result concerns some topological properties of our spaces. Theorem 2 ([7, Theorem (a), (b)]) The sets w p 0 (Λ) and wp (Λ) are BK spaces with respect to the natural norms defined by ( ) /p x Λ = sup ν λ n(ν+) x k p. k K <ν> DUALS Now, we give the β duals of w p 0 (Λ) and wp (Λ). We introduce a few notations. Let a be a sequence and X be a normed sequence space. Then, we write a X = sup x = a kx k provided the expression on the right exists and is finite, which is the case whenever X is a BK space and a X β ([5, Theorem 7.2.9]). If Λ is an exponentially bounded sequence with an associated subsequence, then we write max ν and ν for the maximum and sum taken over all k K <ν>. We denote by x <ν> = ν x k e (k) (ν N 0 ) the ν block of the sequence x. Finally, we write σ( x p )=(σ ν ( x p )) ν=0 and τ( x p )=(τ ν ( x p )) ν=0 for the sequences with σ ν ( x p )= ( λ n(ν+) ) /p x <ν> p and τ ν( x p )=λ /p for ν = 0,,... The duals (w p 0 (Λ))β, (w p (Λ))β and (w p 0 (Λ)) are given the next theorem. n(ν+) x <ν> q Theorem 3 ([6, Theorems 5.5, 5.8; Remark 5.6]) We write { M p (Λ)= a Mp (Λ) < where a Mp (Λ) = τ( a p ). (a) Then, we have Mp (Λ) = w (Λ) p = w p 0 (Λ) on M p(λ);w p 0 (Λ) of w p 0 (Λ) is norm isomorphic to M p(λ) with the norm Mp (Λ); (w (Λ)) p is not given by a sequence space. (b) We have (w (Λ)) p ββ =(w p 0 (Λ))ββ = w (Λ), p M p (Λ) = on (M p(λ)) β, and (M p (Λ)) is norm isomorphic to w p (Λ). VISUALIZATION Finally, we visualize some neighbourhoods of 0 in l(p) and w p (Λ) and M p (Λ). This is done in a natural way as follows

3 FIGURE. S 3 d (p) (0,r) for: Left p =(/2,2,3/2); Right p =(/2,4,/4) FIGURE 2. Left: neighbourhood in the w (Λ) norm p for p=.2, λ =, λ 2 = 6, λ 3 = 9; Right: neighbourhood in the dual norm of the w (Λ) norm p for p=5, λ =, λ 2 = 6, λ 3 = 9 Let P n(k) : ω C for k =,2,3 be coordinates. We represent open balls B r (0) ={x X : d(x,0) < r in a metric sequence space (X,d) by spheres ( 3 ) Sd {(x 3 (0,r)= n(),x n(2),x n(3) ) C 3 : d P n(k) (x)e (n(k)),0 = r. More results on neighbourhoods in FK spaces and their graphical representations can be found in [8]. ACKNOWLEDGMENTS Research of both authors supported by the research project #4F04 of Tübitak, and of the second author also by #74025 of the Serbian Ministry of Science, Technology and Environment

4 REFERENCES. E. Malkowsky, and W. Nickel, Computergrafik in der Differentialgeometrie, Vieweg Verlag, Braunschweig, M. Failing, and E. Malkowsky, Mitt. Math. Sem. Giessen 229, 28 (996). 3. M. Failing, Entwicklung numerischer Algorithmen zur computergrafischen Darstellung spezieller Probleme der Differentialgeometrie und Kristallografie, Ph.D. thesis, Giessen, Shaker Verlag Aachen (996). 4. E. Malkowsky, An open software in OOP for computer graphics and some applications in differential geometry, in Proceedings of the 20th South African Symposium on Numerical Mathematics, 994, pp A. Wilansky, Summability through Functional Analysis, vol. 85, Mathematics Studies, North Holland, Amsterdam, E. Malkowsky, and V. Veličković, Topology and its Applications 58, (20). 7. E. Malkowsky, Acta Sci. Math. (Szeged) 6, (995). 8. E. Malkowsky, F. Özger, and V. Veličković, MATCH Commun. Math. Comput. Chem. 70, (203)

5 AIP Conference Proceedings is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: For more information, see