Abstract
Problem statement: In the theory of Banach spaces one of the problems which describes geometric property of Banach spaces is Banach-Saks Property. In this context we were known many Banach spaces which had this property such as Lp[0, 1] for 1 < p ≤ 2. Approach: Following the sequential structure of the Banach sequence space mp(X), for 1 ≤ p < 8, defined in[1], we arrived to describe a geometric property of this Banach spaces. Results: In this note we showed that Banach spaces mp(X), for 1 ≤ p < 8 had the Banach-Saks Property. Conclusion/Recommendations: Based in present approach, we recommend using our method to study the weak Banach-Saks property in sequential Banach spaces.
Highlights
The Banach-Saks property was studied in Banach spaces and several characterizations were given for it
In this note we prove that the Banach space mp(X), for 1 ≤ p < 8 has the Banach-Saks property
(Alternatively, we may take Λ to be the vector space of complex scalar sequences and what follows remains true in both cases, real and complex)
Summary
The Banach-Saks property was studied in Banach spaces and several characterizations were given for it. In[5], was studied Banach-Saks property in the product of Banach spaces. In this note we prove that the Banach space mp(X), for 1 ≤ p < 8 has the Banach-Saks property. The sequence space mp(X) was defined by[1].
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