Point and Uniformly Point in Orlicz-Bochner Sequence Spaces
2020-07-28 01:13:30WANGKangji王康吉GONGWanzhong鞏萬中
應(yīng)用數(shù)學(xué) 2020年3期
WANG Kangji(王康吉),GONG Wanzhong(鞏萬中)
(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
