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Mat. Sb. (N.S.), 1975, Volume 97(139), Number 2(6), Pages 193–229 (Mi msb3648)  

This article is cited in 35 scientific papers (total in 35 papers)

On a dual problem. I. General results. Applications to Frèchet spaces

Yu. F. Korobeinik

Abstract: Let $H$ be a separated locally convex space; $x_k\in H$, $x_k\ne0$, $k=1,2,…$ . The author shows that if $H$ is a Frèchet space or an $LN^*$-space, then the system $\{x_k\}$ is a basis (topological or absolute) in the closure of its linear span if and only if the system of equations $\varphi(x_k)=d_k$, $k=1,2,…$, has a solution $\varphi$ in $H'$ for any sequence $\{d_k\}$ from a certain space $E_1$ (respectively, from $E_2$ for an absolute basis).
Bibliography: 32 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 26:2, 181–212

Bibliographic databases:

UDC: 517.52+513.88
MSC: Primary 46A35, 46A20; Secondary 46A05, 46E10, 46A45, 30A64, 30A98
Received: 16.04.1974

Citation: Yu. F. Korobeinik, “On a dual problem. I. General results. Applications to Frèchet spaces”, Mat. Sb. (N.S.), 97(139):2(6) (1975), 193–229; Math. USSR-Sb., 26:2 (1975), 181–212

Citation in format AMSBIB
\by Yu.~F.~Korobeinik
\paper On~a~dual problem. I.~General results. Applications to Fr\`echet spaces
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 97(139)
\issue 2(6)
\pages 193--229
\jour Math. USSR-Sb.
\yr 1975
\vol 26
\issue 2
\pages 181--212

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    This publication is cited in the following articles:
    1. Yu. F. Korobeinik, “On a dual problem. II. Applications to $LN^*$-spaces and other questions”, Math. USSR-Sb., 27:1 (1975), 1–22  mathnet  crossref  mathscinet  zmath
    2. A. V. Bratishchev, Yu. F. Korobeinik, “The multiple interpolation problem in the space of entire functions of given proximate order”, Math. USSR-Izv., 10:5 (1976), 1049–1074  mathnet  crossref  mathscinet  zmath
    3. V. A. Oskolkov, “Some bases in spaces of regular functions and their application to interpolation”, Math. USSR-Sb., 34:2 (1978), 215–234  mathnet  crossref  mathscinet  zmath
    4. Yu. F. Korobeinik, “Representing systems”, Math. USSR-Izv., 12:2 (1978), 309–335  mathnet  crossref  mathscinet  zmath
    5. Yu. F. Korobeinik, “The moment problem, interpolation and basicity”, Math. USSR-Izv., 13:2 (1979), 277–306  mathnet  crossref  mathscinet  zmath  isi
    6. A. V. Abanin, “Problem concerning a system of elements being a basis in its linear span”, Funct. Anal. Appl., 13:4 (1979), 278–279  mathnet  crossref  mathscinet  zmath
    7. Yu. F. Korobeinik, “Interpolation problems, nontrivial expansions of zero, and representing systems”, Math. USSR-Izv., 17:2 (1981), 299–337  mathnet  crossref  mathscinet  zmath  isi
    8. V. A. Tkachenko, “Spectral theory in spaces of analytic functionals for operators generated by multiplication by the independent variable”, Math. USSR-Sb., 40:3 (1981), 387–427  mathnet  crossref  mathscinet  zmath  isi
    9. Korobeinik I., “Representing Systems of Exponents and Non-Trivial Decompositions of Zero”, 252, no. 3, 1980, 528–531  mathscinet  isi
    10. Yu. F. Korobeinik, “Representing systems”, Russian Math. Surveys, 36:1 (1981), 75–137  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. Napalkov V., “The Comparison of 2 Topologies in Some Spaces of Entire-Functions”, 264, no. 4, 1982, 827–830  mathscinet  zmath  isi
    12. Yu. F. Korobeinik, “On some representing systems in spaces of analytic functions”, Math. USSR-Izv., 23:3 (1984), 487–509  mathnet  crossref  mathscinet  zmath
    13. A. V. Bratishchev, “A type of lower estimate for entire functions of finite order, and some applications”, Math. USSR-Izv., 24:3 (1985), 415–438  mathnet  crossref  mathscinet  zmath
    14. Korobeinik I., “Canonical Biorthogonal Systems - Applications to the Problems of Basicity and Interpolation”, 280, no. 6, 1985, 1298–1302  mathscinet  isi
    15. B. V. Vinnitskii, “On a description of bases of generalized systems of exponentials”, Math. USSR-Sb., 63:1 (1989), 59–80  mathnet  crossref  mathscinet  zmath
    16. V. A. Oskolkov, “On the completeness and quasipower basis property of systems $ż^nf(\lambda_nz)\}$”, Math. USSR-Sb., 66:2 (1990), 383–392  mathnet  crossref  mathscinet  zmath  isi
    17. Porn C., “Systems of Absolute Representation in Spaces of Holomorphic-Functions”, Studia Math., 94:3 (1989), 193–212  mathscinet  zmath  isi
    18. Yu. F. Korobeinik, “Description of the general form of nontrivial expansions of zero in exponentials. Applications”, Math. USSR-Izv., 39:2 (1992), 1013–1032  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    19. Korobeinik Y., Mikhailov A., “Analytic Solutions of a Cauchy-Problem”, Differ. Equ., 27:3 (1991), 361–366  mathnet  mathscinet  zmath  isi
    20. Kadets V., Korobeinik Y., “Representing and Absolutely Representing Systems”, Studia Math., 102:3 (1992), 217–223  crossref  mathscinet  zmath  isi
    21. M. M. Dragilev, “On subspaces of nuclear spaces”, Sib Math J, 37:3 (1996), 490  mathnet  crossref  mathscinet  zmath  isi
    22. Yu. F. Korobeinik, “On the convergence of series in locally convex spaces”, Russian Math. (Iz. VUZ), 45:8 (2001), 58–68  mathnet  mathscinet  zmath  elib
    23. Yu. F. Korobeinik, “On some spaces of functions infinitely differentiable in an open set of $\mathbf R^p$”, Russian Math. (Iz. VUZ), 49:3 (2005), 39–48  mathnet  mathscinet  zmath  elib
    24. Yu. F. Korobeinik, “Linear transformations of representing and absolutely representing systems”, Russian Math. (Iz. VUZ), 49:9 (2005), 17–26  mathnet  mathscinet  elib
    25. D. A. Abanina, “Absolyutno predstavlyayuschie sistemy eksponent s mnimymi pokazatelyami v prostranstvakh ultradzhetov normalnogo tipa i prodolzhenie funktsii po Uitni”, Vladikavk. matem. zhurn., 7:1 (2005), 3–15  mathnet  mathscinet  elib
    26. Yu. F. Korobeinik, “Absolyutno predstavlyayuschie sistemy eksponent v vesovykh prostranstvakh Freshe analiticheskikh funktsii i analog teoremy Peli — Vinera — Shvartsa”, Vladikavk. matem. zhurn., 7:3 (2005), 38–44  mathnet  mathscinet
    27. M. A. Shubarin, “Usloviya interpolyatsionnosti dlya semeistv prostranstv Freshe”, Vladikavk. matem. zhurn., 9:2 (2007), 54–62  mathnet  mathscinet  elib
    28. Yu. F. Korobeinik, “The absolutely representing families in certain classes of locally convex spaces”, Russian Math. (Iz. VUZ), 53:9 (2009), 20–28  mathnet  crossref  mathscinet  zmath  elib
    29. Yu. F. Korobeinik, “Ob $A$-predstavlyayuschikh sistemakh. I”, Vladikavk. matem. zhurn., 11:1 (2009), 43–53  mathnet  mathscinet
    30. Mikhailov K.A., “Absolyutno predstavlyayuschie sistemy podprostranstv v prostranstvakh probnykh ul'tradifferentsiruemykh funktsii”, Izv. RAN. Severo-Kavkazskii region. Ser.: Estestvennye nauki, 2009, no. 6, 8–10  zmath  elib
    31. A. V. Abanin, Le Hai Khoi, “Pre-dual of the Function Algebra A −∞(D) and Representation of Functions in Dirichlet Series”, Complex anal oper theory, 2010  crossref  mathscinet
    32. V. B. Sherstyukov, “Dual characterization of absolutely representing systems in inductive limits of Banach spaces”, Siberian Math. J., 51:4 (2010), 745–754  mathnet  crossref  mathscinet  isi  elib  elib
    33. D. A. Abanina, “Representation of solutions of convolution equations in nonquasianalytic Beurling classes of ultradifferentiable functions of mean type”, Russian Math. (Iz. VUZ), 55:6 (2011), 1–8  mathnet  crossref  mathscinet  elib
    34. Korobeinik Yu.F., “O nekotorykh klassakh predstavlyayuschikh sistem”, Matematicheskii forum (Itogi nauki. Yug Rossii), 5 (2011), 71–85  elib
    35. Yu. F. Korobeinik, “On Free Representing Systems”, Math. Notes, 91:2 (2012), 198–206  mathnet  crossref  crossref  mathscinet  isi  elib  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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