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Theory of Approximation for Operators in Intuitionistic Fuzzy Normed Linear Spaces
Published in S. A. Mohiuddine, Bipan Hazarika, Sequence Space Theory with Applications, 2023
Nabanita Konwar, Pradip Debnath
The APs take paramount importance in the development of infinite dimensional Banach space theory and also in the study of Schauder basis. A Banach space satisfy the BAPs if and only if it is a complemented subspace of a Banach space with a basis. In this branch of mathematics, the concept of basis implies the Schauder basis. The idea of Schauder basis was introduced by Schauder in 1927. Initially, when Grothendick started to study the approximation property he used the concept of the topology of uniform convergence on compact sets. In 1972, Enflo [15] provide the first example of a complete normed linear space which fails to satisfy the AP. Also there exist some Banach spaces with the BAP which fail to have bases. However, a complemented subspace of Lp for 1<p<∞ has a basis. In 1992, Lusky [30] has established that the Disk Algebra A has a basis without using the results from the complex function theory, but using the results from the AP. Some of the important references which are related to the present work is found in [8, 14, 24, 34, 43].
Re-examination of Bregman functions and new properties of their divergences
Published in Optimization, 2019
Daniel Reem, Simeon Reich, Alvaro De Pierro
We finish this section with the following proposition which describes a sufficient condition for a mapping to be weak-to-weak sequentially continuous, hence helping in establishing examples of functions satisfying Proposition 4.13(XIX). See also Remark 5.7 following this proposition for examples of corresponding Banach spaces satisfying the conditions mentioned in Proposition 5.6. Before formulating this proposition we need a short discussion and a definition. Recall that a Schauder basis of a real infinite-dimensional Banach space is a sequence of elements in X having the property that each can be represented uniquely as a countable linear combination of the basis, namely, for each , there exists a unique sequence of real numbers (the coordinates of x) such that . A standard (algebraic) basis in a finite-dimensional space can also be regarded as a Schauder basis.
The negative exponential transformation: a linear algebraic approach to the Laplace transform
Published in International Journal of Mathematical Education in Science and Technology, 2023
Here we use the term basis to mean a type of Schauder basis where infinite sums are allowed. For students this can be taught as the usual finite sums or polynomial functions approximating the analytic function to any desired proximity (locally). Strictly speaking, a Schauder basis is set in the setting of a Banach function space where approximation is in the norm of the Banach function space.