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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

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