Point and Uniformly Point in Orlicz-Bochner Sequence Spaces

WANG Kangji(王康吉),GONG Wanzhong(巩万中)

(Department of Mathematics,Anhui Normal University,Wuhu 241000,China)

Abstract: Some characterizations of point and uniformly point in Banach space were given.Moreover,in this paper some criteria were obtained forpoint and uniformly point in Orlicz-Bochner sequence spaces.

Key words: point;Uniformly point;Orlicz-Bochner sequence space;Orlicz norm;Luxemburg norm

1.Introduction

2.Some Lemmas

The remainder of the proof shall be given in two cases.

Case 2vm(i0)=0 for somem ∈{2,··· ,n}.Without loss of generality,we may assume that for somek ∈{1,2,··· ,n},vj(i0)≠0 for anyj ∈{1,2,··· ,n−k}andvj(i0)=0 for anyj ∈{n−k+1,··· ,n}.Similarly as Case 1,in view ofM(αu)<αM(u) wheneverα ∈(0,1)andu≠0,we obtain

Therefore by the convexity ofMand the equality above we have