One-complemented subspaces of Musielak–Orlicz sequence spaces

Abstract

The aim of this paper is to characterize one-complemented subspaces of finite codimension in the Musielak–Orlicz sequence space l Φ . We generalize the well-known fact (Ann. Mat. Pura Appl. 152 (1988) 53; Period. Math. Hungar. 22 (1991) 161; Classical Banach Spaces I, Springer, Berlin, 1977) that a subspace of finite codimension in l p , 1 ⩽ p < ∞ , is one-complemented if and only if it can be expressed as a finite intersection of kernels of functionals with at most two coordinates different from zero. Under some smoothness condition on Φ = ( φ n ) we prove a similar characterization in l Φ . In the case of Orlicz spaces we obtain a complete characterization of one-complemented subspaces of finite codimension, which extends and completes the results in Randrianantoanina (Results Math. 33(1–2) (1998) 139). Further, we show that the well-known fact that a one-complemented subspace of finite codimension in l p , 1 ⩽ p < ∞ , is an intersection of one-complemented hyperplanes, is no longer valid in Orlicz or Musielak–Orlicz spaces. In the last section we characterize l p -spaces, 1 < p < ∞ , and separately l 2 -spaces, in terms of one-complemented hyperplanes, in the class of Musielak–Orlicz and Orlicz spaces as well.