Arzelà-Ascoli's theorem in uniform spaces

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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 and 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.
[2]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.
[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.
[4]	I. M. James, Topologies and Uniformities Springer, London, 1999. doi: 10.1007/978-1-4471-3994-2.
[5]	J. L. Kelley, General Topology Springer, Harrisonburg, 1955.
[6]	G. Köthe, Topological Vector Spaces I Springer-Verlag, New York, 1969.
[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.
[8]	V. Maz'ya and S. Poborchi, Differentiable Functions on Bad Domains World Scientific, Singapore, 2001. doi: 10.1142/3197.
[9]	R. Meise and D. Vogt, Introduction to Functional Analysis Oxford: Clarendon Press, Oxford, 1997.
[10]	J. Munkres, Topology Prentice Hall, Upper Saddle River, 2000.
[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.
[12]	W. Rudin, Functional Analysis McGraw-Hill Inc., Singapore, 1991.
[13]	L. Schwartz, Mathematics for the Physical Sciences Addison-Wesley Publishing Company, Paris, 1966.
[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.
[15]	R. Strichartz, A Guide to Distribution Theory and Fourier Transforms CRC Press, Boca Raton, 1994.
[16]	S. Willard, General Topology Addison-Wesley Publishing Company, Reading, 1970.
[17]	K. Yosida, Functional Analysis Springer-Verlag, Berlin, 1980.

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.
[2]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.
[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.
[4]	I. M. James, Topologies and Uniformities Springer, London, 1999. doi: 10.1007/978-1-4471-3994-2.
[5]	J. L. Kelley, General Topology Springer, Harrisonburg, 1955.
[6]	G. Köthe, Topological Vector Spaces I Springer-Verlag, New York, 1969.
[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.
[8]	V. Maz'ya and S. Poborchi, Differentiable Functions on Bad Domains World Scientific, Singapore, 2001. doi: 10.1142/3197.
[9]	R. Meise and D. Vogt, Introduction to Functional Analysis Oxford: Clarendon Press, Oxford, 1997.
[10]	J. Munkres, Topology Prentice Hall, Upper Saddle River, 2000.
[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.
[12]	W. Rudin, Functional Analysis McGraw-Hill Inc., Singapore, 1991.
[13]	L. Schwartz, Mathematics for the Physical Sciences Addison-Wesley Publishing Company, Paris, 1966.
[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.
[15]	R. Strichartz, A Guide to Distribution Theory and Fourier Transforms CRC Press, Boca Raton, 1994.
[16]	S. Willard, General Topology Addison-Wesley Publishing Company, Reading, 1970.
[17]	K. Yosida, Functional Analysis Springer-Verlag, Berlin, 1980.

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