Mathematics and Statistics Vol. 11(4), pp. 661 - 668
DOI: 10.13189/ms.2023.110407
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Some Results on Sequences in Banach Spaces

B. M. Cerna Maguina ^1,*, Miguel A. Tarazona Giraldo ^2
1 Academic Department of Mathematics, Science Faculty, Santiago Antúnez de Mayolo National University, Shancayan Campus, Peru
² Faculty of Electronic and Computer Engineering, Federico Villarreal National University, Peru

ABSTRACT

In this work, we prove in a very particular way the theorems of Dvoretzky-Roger's, Shur's, Orcliz's and Theorem 14.2 in their versions presented in the text [3]. The demonstrations of these Theorems carried out by us consist in establishing an appropriate link between the object of study and the relation that affirms that, for any real numbers , there exists a unique real number such that . Once the nexus is established, we use the definition of weak or strong convergence together with the Hahn-Banach Theorem to obtain the desired results. The relation is obtained by decomposing the Hilbert space as the direct sum of a closed subspace and its orthogonal complement. Since the dimension of the space is finite, this guarantees that any linear functional defined on the space is continuous, and this guarantees that the kernel of said linear functional is closed in the space . Therefore we have that the space breaks down, as the direct sum of the kernel of the continuous linear functional and its orthogonal complement, that is: , where the dimension of ker and the dimension of .

KEYWORDS
Functional Analysis, Numerical Analysis, Dvoretzky-Rogers Theorem, Orlicz's Theorem, Shur's Theorem

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] B. M. Cerna Maguina , Miguel A. Tarazona Giraldo , "Some Results on Sequences in Banach Spaces," Mathematics and Statistics, Vol. 11, No. 4, pp. 661 - 668, 2023. DOI: 10.13189/ms.2023.110407.

(b). APA Format:
B. M. Cerna Maguina , Miguel A. Tarazona Giraldo (2023). Some Results on Sequences in Banach Spaces. Mathematics and Statistics, 11(4), 661 - 668. DOI: 10.13189/ms.2023.110407.