Copies of $l_{\infty}$ in Köthe spaces of vector valued functions

Abstract

Let (S, E, x) be a r-finite complete measure space and X be a Banach space.Recently the following result has appeared THEOREM 1 [71.if it embeds into X.Let 1 < p < .Then 1 embeds into L P(Iz, X) if and onlyThe purpose of this note is to extend Theorem 1 to a more general class of vector-valued functions; namely, K6the spaces E(X)of vector-valued func- tions.Specifically, we show that l embeds into E(X) if and only if it embeds into either E or X.We recall that L P(tz, X) spaces as well as Orlicz or Musielak-Orlicz spaces of vector-valued functions are special cases of K6the spaces.Before giving our result, we need some definitions and results.Let /(S) / be the space of E-measurable real valued functions with func- tions equal -almost everywhere identified.A K6the space E [6] is a Banach subspace of ' consisting of locally integrable functions such that (i) if[u[ _< v[ /z.a.e., with u /, v E then u E and [[ulle -< Ilvlle, (ii) for each A E,/z(A) < o% the characteristic function X of A is in E. K6the spaces are Banach lattices if we put u >_ 0 when u(s) >_ 0 Iz.a.e.Further- more, K6the spaces are r-complete Banach lattices.The following theorems will be utilized in the sequel.THEOREM 2 [5].Given a K6the space E, there exists an increasing sequence (S n) in with I(Sn) < and I(S \ [.J n NS) 0 for which the following chain of continuous inclusions holds" L=( S,) c E( S) L'(

Locations

• Illinois Journal of Mathematics - View - PDF