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Weak sequential completeness of \(\beta\)-duals. (English) Zbl 0885.46007

Let \(\omega\) be the vector space of all (real or complex) sequences and \(\varphi\) its vector subspace of all finite sequences. A sequence \((k_v)\) of natural numbers is said to be an index sequence if \(k_v< k_{v+1}\) \((v\in\mathbb{N})\). A sequence \((y^{(n)})\), \(n\in\mathbb{N}\), of non-zero sequences from \(\omega\) is a block sequence if there exists an index sequence \((k_j)\), \(j\in\mathbb{N}\), such that \[ y^{(n)}= (0,0,\dots, 0,y^{(n)}_{k_{n-1}+ 1},\dots, y^{(n)}_{k_n}, 0,0,\dots)= \sum^{k_n}_{j= k_{n-1}+1} y^{(n)}_j e^{(j)}, \] where \(e^{(j)}= (0,0,\dots, 0,1,0,\dots)\) with \(1\) in the \(j\)th place.
In his paper “Sequential completeness and spaces with the gliding humps property” [Manuscr. Math. 66, No. 3, 237-252 (1990; Zbl 0713.46009)], D. Noll introduced the notion of the weak gliding humps property for a sequence space \(X\) containing \(\varphi\). \(X\) is said to have the weak gliding humps property (WGHP) if, given any \(x\in X\) and any block sequence \((x^{(k)})\) such that \(x= \sum_i x^{(i)}\) (pointwise sum), every sequence \((n_k)\) of integers admits a subsequence \((m_k)\) such that \(\widetilde x=\sum_k x^{(m_k)}\in X\) (pointwise sum). In the paper mentioned above, D. Noll proved that the \(\beta\)-dual \(X^\beta\) of a sequence space \(X\) which contains \(\varphi\) and has the WGHP is \(\sigma(X^\beta, X)\) sequentially complete, i.e. every \(\sigma(X^\beta,X)\)-Cauchy sequence converges to an element of \(X^\beta\). In the present paper, the author considers real sequence spaces and works on the following program:
At first he states and proves, in the setting of Abelian topological groups, a generalization of the basic matrix theorem of Antosik and Mikusinski (Theorem 3.2). Then he generalizes Noll’s definition of WGHP by introducing the notion of the signed weak gliding hump property (signed WGHP). A sequence space \(X\) containing \(\varphi\) has the signed WGHP if, given any \(x\in X\) and any disjoint sequence \((I_n)\) of finite subintervals of \(\mathbb{N}\), there exists a subsequence \((I_{n(k)})\) and a choice of signs \((s_k)\in \{-1,1\}^{\mathbb{N}}\) such that the coordinatewise sum \(\sum_k s_kC_{I_{n(k)}}x\in X\). (\(C_A\) denotes the characteristic function of \(A\)). The author remarks that many sequence spaces have the signed WGHP and gives the example of the sequence space \[ bs= \Biggl\{(x_j): \sup_n\Biggl|\sum_{j\leq n} x_j\Biggr|< \infty\Biggr\} \] which has the signed WGHP but not the WGHP.
Finally, using Theorem 3.2., the author states and proves his main result in the paper which is the following generalization of the above-mentioned Noll’s result.
Let \(E\) be a sequence space containing \(\varphi\) and having the signed WGHP. Then \(E^\beta\) is \(\sigma(E^\beta, E)\) sequentially complete (Theorem 3.5.).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46A35 Summability and bases in topological vector spaces
40D25 Inclusion and equivalence theorems in summability theory
40H05 Functional analytic methods in summability

Citations:

Zbl 0713.46009
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References:

[1] P. Antosik and C. Swartz, Matrix methods in analysis , Springer-Verlag, Heidelberg, 1985. · Zbl 0564.46001 · doi:10.1007/BFb0072264
[2] G. Bennett, A new class of sequence spaces with applications in summability theory , J. Reine Angew Math. 266 (1974), 49-75. · Zbl 0277.46012 · doi:10.1515/crll.1974.266.49
[3] G. Bennett and N.J. Kalton, Inclusion theorems for \(K\)-spaces , Canad. J. Math. 25 (1973), 511-524. · Zbl 0272.46009 · doi:10.4153/CJM-1973-052-2
[4] ——–, Consistency theorems for almost convergence , Trans. Amer. Math. Soc. 198 (1974), 23-43. · Zbl 0301.46005 · doi:10.2307/1996745
[5] J. Boos and D. Fleming, Gliding hump properties and some applications , · Zbl 0824.46007 · doi:10.1155/S0161171295000160
[6] J. Boos and T. Leiger, General theorems of Mazur-Orlicz type , Stud. Math. 42 (1989), 1-19. · Zbl 0703.46004
[7] D.J.H. Garling, The \(\b\)- and \(\g\)-duality of sequence spaces , Proc. Cambridge Philos. Soc. 63 (1967), 963-981. · Zbl 0161.10401 · doi:10.1017/S0305004100041992
[8] P.K. Kamthan and M. Gupta, Sequence spaces and series , Marcel Dekker, New York, 1981. · Zbl 0447.46002
[9] D. Noll, Sequential completeness and spaces with the gliding humps property , Manuscripta Math. 66 (1990), 237-252. · Zbl 0713.46009 · doi:10.1007/BF02568494
[10] A.K. Snyder, Consistency theory and semiconservative spaces , Stud. Math. 71 (1982), 1-13. · Zbl 0524.46003
[11] C. Swartz, Weak sequential completeness of sequence spaces , Collect. Math., · Zbl 0782.46013
[12] ——–, The gliding hump property in vector sequence spaces , · Zbl 0798.46003 · doi:10.1007/BF01404009
[13] ——–, An introduction to functional analysis , Marcel Dekker, 1992. · Zbl 0751.46002
[14] J. Swetits, A characterization of a class of barrelled sequence spaces , Glasgow Math. J. 19 (1978), 27-31. · Zbl 0401.46004 · doi:10.1017/S0017089500003335
[15] ——–, On the relationship between a summability matrix and its transpose , J. Austral. Math. Soc. (1980), 362-368. · Zbl 0441.40008 · doi:10.1017/S1446788700012325
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