China
Africa
Explore chapters and articles related to this topic
Banach Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
The third fundamental theorem is the uniform boundedness principle (Theorem 3.7.4 for pointwise bounded sequences of continuous linear functionals on a Fréchet space). It asserts that any set of pointwise bounded continuous linear maps from one Banach space to another is uniformly bounded on the unit ball. The uniform boundedness principle was proved by Hahn in 1922 for continuous linear functionals on a Banach space. Later it was proved by Hildebrandt for continuous linear maps between Banach spaces in 1923 and also proved in a more general version by Banach and Steinhaus in 1927. The uniform boundedness principle is also referred to as the Banach-Steinhaus theorem (Theorem 3.7.5 for pointwise convergent sequences of continuous linear functionals on a Fréchet).
Fuzzifying topology induced by Morsi fuzzy pseudo-norms
Published in International Journal of General Systems, 2022
The aim of this study is to fill in the blanks on the research of fuzzy normed vector space defined by Morsi. At first, the relationships between the family of ordinary pseudo-norms and the fuzzy pseudo-norm in the sense of Morsi are studied. Then a fuzzifying topology on fuzzy pseudo-normed spaces is introduced, and this fuzzifying topology is compatible with the structure of vector operation. The definition of the degree to which a set is fuzzy bounded by means of fuzzy norms is given, and it is proved that the degree to which a set is fuzzy bounded by means of fuzzy norms is equal to the degree of a set with the help of fuzzifying topologies. The characterizations of fuzzy continuous linear operators and fuzzy bounded operators in terms of layered structure are obtained. At last, the definition of the degree to which a sequence is convergent a point is presented, and the conclusion which the uniqueness of the limit of convergence point series is proved. Based on the many-valued topological structure, we establish a fundamental framework of fuzzy normed vector spaces. It provides a theoretical basis for further study of fuzzy normed vector space using many-valued topology. Based on this framework, we can develop the theory of bounded linear operators systematically, and establish some fuzzy versions of the open mapping theorem and the uniform boundedness principle; we can discuss the stability of mixed type functional equation in Morsi's fuzzy normed spaces as Narasimman, Dutta, and Jebril (2019), we can consider their applications in the theory of fixed points and other nonlinear problems. This study also provides a model for the regularity theory in fuzzy topological space.