Lower Bound of the Upper Topological Limit of a Sequence of Subspaces

Aram V. Arutyunov

doi:10.68381/jca27004

Abstract

A new short proof of the following assertion is presented. Given an integer k and a sequence of closed linear subspaces of a Banach space, assume that the codimension of each subspace does not exceed k. Then the upper topological limit of this sequence contains a closed linear subspace with codimension not exceeding k.

Author information

Contact details are reproduced from the original publication and may be historical.

Aram V. Arutyunov

V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

arutyunov@cs.msu.ru

Keywords

Sequence of subspaces
upper topological limit
lower bound

Mathematics Subject Classification

46B20

References

A. V. Arutyunov: Second-order conditions in extremal problems. The abnormal points, Trans. Amer. Math. Soc. 350 (1998) 4341–4365.https://doi.org/10.1090/S0002-9947-98-01775-9 Google Scholar
A. V. Arutyunov: Optimality conditions: Abnormal and degenerate problems, Mathematics and its Applications 526, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-94-015-9438-7 Google Scholar
A. V. Arutyunov: Smooth abnormal problems in extremum theory and analysis, Russ. Math. Surveys 67 (2012) 403–457.https://doi.org/10.1070/RM2012v067n03ABEH004793 Google Scholar
V. I. Bogachev, O. G. Smolyanov: Real and Functional Analysis: a University Course, in: Regular and Chaotic Dynamics, M.-Izhevsk (ed.), Moscow (2009).Google Scholar
W. Rudin: Functional Analysis, McGraw-Hill, New York (1973).Google Scholar
K. V. Storozhuk: On the upper topological limit of a family of vector subspaces of codimension k, Sib. Electron. Math. Izv. 12 (2015) 432–435.Google Scholar

Suggested citation

Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.

[1] A. V. Arutyunov: Second-order conditions in extremal problems. The abnormal points, Trans. Amer. Math. Soc. 350 (1998) 4341–4365.https://doi.org/10.1090/S0002-9947-98-01775-9 Google Scholar

[2] A. V. Arutyunov: Optimality conditions: Abnormal and degenerate problems, Mathematics and its Applications 526, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-94-015-9438-7 Google Scholar

[3] A. V. Arutyunov: Smooth abnormal problems in extremum theory and analysis, Russ. Math. Surveys 67 (2012) 403–457.https://doi.org/10.1070/RM2012v067n03ABEH004793 Google Scholar

[4] V. I. Bogachev, O. G. Smolyanov: Real and Functional Analysis: a University Course, in: Regular and Chaotic Dynamics, M.-Izhevsk (ed.), Moscow (2009).Google Scholar

[5] W. Rudin: Functional Analysis, McGraw-Hill, New York (1973).Google Scholar

[6] K. V. Storozhuk: On the upper topological limit of a family of vector subspaces of codimension k, Sib. Electron. Math. Izv. 12 (2015) 432–435.Google Scholar