Absolute boundedness and absolute convergence in sequence spaces
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- by Martin Buntinas and Naza Tanović-Miller PDF
- Proc. Amer. Math. Soc. 111 (1991), 967-979 Request permission
Abstract:
Let $\mathcal {H}$ be the set of all sequences $h = ({h_k})_{k = 1}^\infty$ of 0s and 1s. A sequence $x$ in a topological sequence space $E$ has the property of absolute boundedness $|AB|$ if $\mathcal {H}\cdot x = \{ y|{y_k} = {h_k}{x_k},h \in \mathcal {H}\}$ is a bounded subset of $E$. The subspace ${E_{\left | {AB} \right |}}$ of all sequences with absolute boundedness in $E$ has a natural topology stronger than that induced by $E$. A sequence $x$ has the property of absolute sectional convergence $|AK|$ if, under this stronger topology, the net $\{ h\cdot x\}$ converges to $x$, where $h$ ranges over all sequences in $\mathcal {H}$ with a finite number of 1s ordered coordinatewise $(h’ \leq h''\;{\text {iff}}\;\forall k,{h’_k} \leq {h''_k})$. Absolute boundedness and absolute convergence are investigated. It is shown that, for an $FK$-space $E$, we have $E = {E_{|AB|}}$ if and only if $E = {l^\infty }\cdot E$, and every element of $E$ has the property $|AK|$ if and only if $E = {c_0}\cdot E$. Solid hulls and largest solid subspaces of sequence spaces are also considered. The results are applied to standard sequence spaces, convergence fields of matrix methods, classical Banach spaces of Fourier series and to more recently introduced spaces of absolutely and strongly convergent Fourier series.References
- J. M. Anderson and A. L. Shields, Coefficient multipliers of Bloch functions, Trans. Amer. Math. Soc. 224 (1976), no. 2, 255–265 (1977). MR 419769, DOI 10.1090/S0002-9947-1976-0419769-6 N. Bary, A treatise on trigonometric series, vols. 1 and 2, Pergamon Press, New York, 1964.
- G. Bennett and N. J. Kalton, Inclusion theorems for $K$-spaces, Canadian J. Math. 25 (1973), 511–524. MR 322474, DOI 10.4153/CJM-1973-052-2
- Martin Buntinas, Convergent and bounded Cesàro sections in $\textrm {FK}$-spaces, Math. Z. 121 (1971), 191–200. MR 295020, DOI 10.1007/BF01111591
- Martin Buntinas, On Toeplitz sections in sequence spaces, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 3, 451–460. MR 410163, DOI 10.1017/S0305004100051926
- Martin Buntinas, Products of sequence spaces, Analysis 7 (1987), no. 3-4, 293–304. MR 928643, DOI 10.1524/anly.1987.7.34.293
- Martin Buntinas and Günther Goes, Products of sequence spaces and multipliers, Rad. Mat. 3 (1987), no. 2, 287–300 (English, with Serbo-Croatian summary). MR 931985
- D. J. H. Garling, The $\beta$- and $\gamma$-duality of sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 963–981. MR 218881, DOI 10.1017/s0305004100041992
- D. J. H. Garling, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 997–1019. MR 218880, DOI 10.1017/s0305004100042031
- Casper Goffman and George Pedrick, First course in functional analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0184064
- C. N. Kellogg, An extension of the Hausdorff-Young theorem, Michigan Math. J. 18 (1971), 121–127. MR 280995
- J. J. Sember, On unconditional section boundedness in sequence spaces, Rocky Mountain J. Math. 7 (1977), no. 4, 699–706. MR 448028, DOI 10.1216/RMJ-1977-7-4-699
- John Sember and Marc Raphael, The unrestricted section properties of sequences, Canadian J. Math. 31 (1979), no. 2, 331–336. MR 528812, DOI 10.4153/CJM-1979-036-1
- I. Szalay and N. Tanović-Miller, On Banach spaces of absolutely and strongly convergent Fourier series, Acta Math. Hungar. 55 (1990), no. 1-2, 149–160. MR 1077070, DOI 10.1007/BF01951398
- I. Szalay and N. Tanović-Miller, On Banach spaces of absolutely and strongly convergent Fourier series. II, Acta Math. Hungar. 57 (1991), no. 1-2, 137–149. MR 1128850, DOI 10.1007/BF01903812
- N. Tanović-Miller, On strong convergence of trigonometric and Fourier series, Acta Math. Hungar. 42 (1983), no. 1-2, 35–43. MR 716552, DOI 10.1007/BF01960550
- N. Tanović-Miller, On a paper of Bojanić and Stanojević, Rend. Circ. Mat. Palermo (2) 34 (1985), no. 2, 310–324. MR 814044, DOI 10.1007/BF02850704
- N. Tanović-Miller, Strongly convergent trigonometric series as Fourier series, Acta Math. Hungar. 47 (1986), no. 1-2, 127–135. MR 836403, DOI 10.1007/BF01949133
- N. Tanović-Miller, On Banach spaces of strongly convergent trigonometric series, J. Math. Anal. Appl. 146 (1990), no. 1, 110–127. MR 1041205, DOI 10.1016/0022-247X(90)90336-E
- Karl Zeller, Allgemeine Eigenschaften von Limitierungsverfahren, Math. Z. 53 (1951), 463–487 (German). MR 39824, DOI 10.1007/BF01175646
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 967-979
- MSC: Primary 40H05; Secondary 42A16, 46A45
- DOI: https://doi.org/10.1090/S0002-9939-1991-1039252-0
- MathSciNet review: 1039252