The uniform boundedness theorem in b-Banach space

Volume 6, Issue 2, April 2021     |     PP. 25-32      |     PDF (192 K)    |     Pub. Date: May 17, 2021
DOI: 10.54647/mathematics11256    110 Downloads     5463 Views  

Author(s)

Jiachen Lv, School of Science, Wuhan University of Science and Technology, Wuhan, China
Yuqiang Feng, School of Science, Wuhan University of Science and Technology, Wuhan, China

Abstract
B-Banach space is an extension of Banach space, which provides a suitable framework for studying many analytical problems. The uniform boundedness theorem is is the basic theorem in functional analysis and has many important applications in many field, such as matrix analysis, operator theory, and numerical analysis. In this note, we revisit the concept of b-Banach space, and then establish the uniform boundedness theorem for linear operators. The result may be useful to establish linear operator theory in b-Banach space.

Keywords
B-normed linear space, B-Banach space, Uniform boundedness theorem

Cite this paper
Jiachen Lv, Yuqiang Feng, The uniform boundedness theorem in b-Banach space , SCIREA Journal of Mathematics. Volume 6, Issue 2, April 2021 | PP. 25-32. 10.54647/mathematics11256

References

[ 1 ] I.A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal. Unianowsk Gos. Ped. Inst. 30(1989) 26-37.
[ 2 ] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Univ. Modena 46(1998) 263-276.
[ 3 ] V. Berinde, Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory Preprint no. 3(1993) 3-9.
[ 4 ] V. Berinde, Sequences of operators and fixed points in quasimetric spaces, Stud. Univ. Babes- Bolyai Math. 16(4)(1996) 23-27.
[ 5 ] V. Berinde, Contractii generalizate si aplicatii, Editura Club Press 22, Baia Mare, 1997.
[ 6 ] I. A.Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory 9(2)(2008) 541-559.
[ 7 ] Z. Xu, X. Chang, F. Xu, et al. regularization: a thresholding representation theory and a fast solver, IEEE Trans. Neural Networks Learning Sys., 23(7)(2012), 1013-1027.
[ 8 ] K V Runovskiˇi, On families of linear polynomal operators in -spaces, , Russian Academy of Sciences. Sbornik Mathematics,78(1)(1994)165
[ 9 ] M. Bota, VA. Ilea, A. Petrusel. Krasnoselskii’s theorem in generalized b-Banach spaces and applications, J. Nonlinear Conv. Anal. 18(4) (2017), 575-587.
[ 10 ] M. Bota, A. Karapinar. Fixed point problem under a finite number of equality constraints on b-Banach spaces, Filomat 33(18)(2019), 5837-5849.
[ 11 ] L. Zofia, Banach-Steinhaus theorems for bounded linear operators with values in a generalized 2-normed space. Glas. Mat. Ser. III 38(58) (2003), 329-340.
[ 12 ] Alan D. Sokal, A really simple elementary proof of the uniform boundedness theorem, Amer. Math. Monthly 118(2011), 450-452.
[ 13 ] Mahlon M. Day, The spaces with . Bull. Amer. Math. Soc. 46(1940), 816-823.