A note on near smoothness of Banach spaces

J.Banas

Rzeszów University of Technology, W. Pola 2, 35-959 Rzeszów Poland

F.S.Cabrera, K.Sadarangani

Universidad de Las Palmas de Gran Canaria

Edificio de Informática y Matemáticas, Campus de Tafira, 35.017

Las Palmas de Gran Canaria, Spain

e-mail: fcabrera@dma.ulpgc.es

In the classical geometry of Banach spaces the notion of smoothness plays a very important and fundamental role. This notion finds also numerous applications in other branches of nonlinear functional analysis and control theory, among others. In recent years the notion in question has been generalized in terms of compactness conditions, being the main tool the Hausdorff measure of noncompactness.

An important fact in these generalizations is that the class of the nearly uniformly smooth spaces contains nonreflexive spaces. Particularly, it is proved that the classical spaces is nearly uniformly smooth. Also it is proved that the near smoothness is hereditary by passing to the sequence Banach spaces and .

In this paper it is proved that the notion of near smoothness is not hereditary in -product of a finite number of Banach spaces. Particularly, we prove that the product space under the norm is not a near smooth space. This result proves that the norm has a distinct behaviour that the norms in the sequence Banach spaces under the assumption that the spaces are near smooth spaces.