In the classical geometry of Banach spaces the notions of smoothness, uniform smoothness, strict and uniform convexity introduced by Day and Clarkson play a very important role and are used in many branches of functional analysis. In recent years a lot of papers have appeared containing interesting generalizations of these notions in terms of a measure of noncompactness. These new concepts investigated in this paper as near uniform smoothness, local near uniform smoothness and modulus of near smoothness have been introduced by Stachura and Sȩkowski and Banás.
The main aim of this paper is to provide an estimate of the modulus of near smoothness of the so-called \(\ell^p\) product of a sequence of Banach spaces. Further, we prove that the notions of local near uniform smoothness and convexity are hereditary in an \(\ell^p\) product of spaces. Apart from that we calculate the exact formula for the modulus of near smoothness for the space \(\ell^p(\ell^{p_i})\).