American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Krasnosel'skii formula for parabolic differential inclusions with state constraints
  • DCDS-B Home
  • This Issue
  • Next Article
    Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals
January  2018, 23(1): 283-294. doi: 10.3934/dcdsb.2018020

Arzelà-Ascoli's theorem in uniform spaces

Mateusz Krukowski 1,,

1. 

Łódź University of Technology, Institute of Mathematics, Wólczańska 215, 90-924 Łódź, Poland

* Corresponding author: Mateusz Krukowski

Received  July 2016 Revised  September 2016 Published  January 2018

In the paper, we generalize the Arzelà-Ascoli's theorem in the setting of uniform spaces. At first, we recall the Arzelà-Ascoli theorem for functions with locally compact domains and images in uniform spaces, coming from monographs of Kelley and Willard. The main part of the paper introduces the notion of the extension property which, similarly as equicontinuity, equates different topologies on $C(X,Y)$. This property enables us to prove the Arzelà-Ascoli's theorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki's theorem, we establish the relative compactness of subfamily of $C({\mathbb{R}},{\mathcal{D}}'({\mathbb{R}}^n))$.

Keywords: Arzelà-Ascoli's theorem, uniform spaces, uniformity of uniform convergence (on compacta).
Mathematics Subject Classification: Primary: 46E10; Secondary: 54E15.
Citation: Mateusz Krukowski. Arzelà-Ascoli's theorem in uniform spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 283-294. doi: 10.3934/dcdsb.2018020
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide Springer, Berlin, 1999. doi: 10.1007/978-3-662-03961-8.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.  Google Scholar

[3]

J. J. Duistermaat and J. A. C. Kolk, Distributions. Theory and Applications Birkhäuser, New York, 2010. doi: 10.1007/978-0-8176-4675-2.  Google Scholar

[4]

I. M. James, Topologies and Uniformities Springer, London, 1999. doi: 10.1007/978-1-4471-3994-2.  Google Scholar

[5]

J. L. Kelley, General Topology Springer, Harrisonburg, 1955. Google Scholar

[6]

G. Köthe, Topological Vector Spaces I Springer-Verlag, New York, 1969.  Google Scholar

[7]

M. Krukowski and B. Przeradzki, Compactness result and its applications in integral equations, J. Appl. Anal., 22 (2016), 153-161, arXiv: 1505.02533. doi: 10.1515/jaa-2016-0016.  Google Scholar

[8]

V. Maz'ya and S. Poborchi, Differentiable Functions on Bad Domains World Scientific, Singapore, 2001. doi: 10.1142/3197.  Google Scholar

[9]

R. Meise and D. Vogt, Introduction to Functional Analysis Oxford: Clarendon Press, Oxford, 1997.  Google Scholar

[10]

J. Munkres, Topology Prentice Hall, Upper Saddle River, 2000. Google Scholar

[11]

B. Przeradzki, The existence of bounded solutions for differential equations in Hilbert spaces, Annales Polonici Mathematici, 56 (1992), 103-121.  doi: 10.4064/ap-56-2-103-121.  Google Scholar

[12]

W. Rudin, Functional Analysis McGraw-Hill Inc., Singapore, 1991.  Google Scholar

[13]

L. Schwartz, Mathematics for the Physical Sciences Addison-Wesley Publishing Company, Paris, 1966.  Google Scholar

[14]

R. Stańczy, Hammerstein equation with an integral over noncompact domain, Annales Polonici Mathematici, 69 (1998), 49-60.  doi: 10.4064/ap-69-1-49-60.  Google Scholar

[15]

R. Strichartz, A Guide to Distribution Theory and Fourier Transforms CRC Press, Boca Raton, 1994.  Google Scholar

[16]

S. Willard, General Topology Addison-Wesley Publishing Company, Reading, 1970.  Google Scholar

[17]

K. Yosida, Functional Analysis Springer-Verlag, Berlin, 1980.  Google Scholar

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide Springer, Berlin, 1999. doi: 10.1007/978-3-662-03961-8.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.  Google Scholar

[3]

J. J. Duistermaat and J. A. C. Kolk, Distributions. Theory and Applications Birkhäuser, New York, 2010. doi: 10.1007/978-0-8176-4675-2.  Google Scholar

[4]

I. M. James, Topologies and Uniformities Springer, London, 1999. doi: 10.1007/978-1-4471-3994-2.  Google Scholar

[5]

J. L. Kelley, General Topology Springer, Harrisonburg, 1955. Google Scholar

[6]

G. Köthe, Topological Vector Spaces I Springer-Verlag, New York, 1969.  Google Scholar

[7]

M. Krukowski and B. Przeradzki, Compactness result and its applications in integral equations, J. Appl. Anal., 22 (2016), 153-161, arXiv: 1505.02533. doi: 10.1515/jaa-2016-0016.  Google Scholar

[8]

V. Maz'ya and S. Poborchi, Differentiable Functions on Bad Domains World Scientific, Singapore, 2001. doi: 10.1142/3197.  Google Scholar

[9]

R. Meise and D. Vogt, Introduction to Functional Analysis Oxford: Clarendon Press, Oxford, 1997.  Google Scholar

[10]

J. Munkres, Topology Prentice Hall, Upper Saddle River, 2000. Google Scholar

[11]

B. Przeradzki, The existence of bounded solutions for differential equations in Hilbert spaces, Annales Polonici Mathematici, 56 (1992), 103-121.  doi: 10.4064/ap-56-2-103-121.  Google Scholar

[12]

W. Rudin, Functional Analysis McGraw-Hill Inc., Singapore, 1991.  Google Scholar

[13]

L. Schwartz, Mathematics for the Physical Sciences Addison-Wesley Publishing Company, Paris, 1966.  Google Scholar

[14]

R. Stańczy, Hammerstein equation with an integral over noncompact domain, Annales Polonici Mathematici, 69 (1998), 49-60.  doi: 10.4064/ap-69-1-49-60.  Google Scholar

[15]

R. Strichartz, A Guide to Distribution Theory and Fourier Transforms CRC Press, Boca Raton, 1994.  Google Scholar

[16]

S. Willard, General Topology Addison-Wesley Publishing Company, Reading, 1970.  Google Scholar

[17]

K. Yosida, Functional Analysis Springer-Verlag, Berlin, 1980.  Google Scholar

[1]

Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477
[2]

Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080
[3]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109
[4]

Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025
[5]

Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873
[6]

Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613
[7]

Gaocheng Yue, Chengkui Zhong. Global attractors for the Gray-Scott equations in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 337-356. doi: 10.3934/dcdsb.2016.21.337
[8]

Surabhi Tiwari, Pankaj Kumar Singh. Rough semi-uniform spaces and its image proximities. Electronic Research Archive, 2020, 28 (2) : 1095-1106. doi: 10.3934/era.2020060
[9]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547
[10]

Tanja Eisner, Pavel Zorin-Kranich. Uniformity in the Wiener-Wintner theorem for nilsequences. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3497-3516. doi: 10.3934/dcds.2013.33.3497
[11]

Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227
[12]

Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062
[13]

Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044
[14]

Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049
[15]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
[16]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
[17]

Martin Swaczyna, Petr Volný. Uniform motions in central fields. Journal of Geometric Mechanics, 2017, 9 (1) : 91-130. doi: 10.3934/jgm.2017004
[18]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048
[19]

Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134
[20]

Lan Wen. A uniform $C^1$ connecting lemma. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 257-265. doi: 10.3934/dcds.2002.8.257

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (82)
  • HTML views (120)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences