Some Results on Sequences in Banach Spaces

Cerna Maguiña B.M.; Tarazona Giraldo M.A. · Mathematics and Statistics (2023) · DOI: 10.13189/ms.2023.110407

Resumen

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 n real numbers a1, a2, …, an, there exists a unique real number λ such that∑n k=1ak = λ∑n k=1a2k. 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∑n k=1ak = λ(∑n k=1 k)a2is obtained by decomposing the Hilbert space Rn as the direct sum of a closed subspace and its orthogonal complement. Since the dimension of the space Rn is finite, this guarantees that any linear functional defined on the space Rn is continuous, and this guarantees that the kernel of said linear functional is closed in the space Rn . Therefore we have that the space Rn breaks down, as the direct sum of the kernel of the continuous linear functional f and its orthogonal complement, that is: Rn = ker f ⊕ [ker f]⊥, where the dimension of ker f = n − 1 and the dimension of [ker f]⊥ = 1.

Instituciones: Universidad Nacional Federico Villarreal; Universidad Nacional Santiago Antúnez de Mayolo