On the Finite Dimensionality of Closed Subspaces in Lp(M,dμ)∩Lq(M,dν)

Finding effective finite-dimensional criteria for closed subspaces in Lp, endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces, Sp⊂Lp, whose finite dimensionality is not fixed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace Sp⊂Lp is constructed by means of some additional conditions and constraints on Lp with no direct exemplification of the functional structure of its elements. We consider a closed topological subspace, Sp(q), of the functional Banach space, Lp(M,dμ), and, moreover, one assumes that additionally, Sp(q)⊂Lq(M,dν) is subject to a probability measure ν on M. Then, we show that closed subspaces of Lp(M,dμ)∩Lq(M,dν) for q>max{1,p},p>0 are finite dimensional. The finite dimensionality result concerning the case when q>p>0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to Lp(M,dμ)∩Lq(M,dν).

The fit and flat components of barrelled spaces

The Splitting Theorem says that any given Hamel basis for a (Hausdorff) barrelled space E determines topologically complementary subspaces Ec and ED, and that Ec is flat, that is, contains no proper dense subspace. By use of this device it was shown that if E is non-flat it must contain a dense subspace of codimension at least ℵ0; here we maximally increase the estimate to ℵ1 under the assumption that the dominating cardinal ∂ equals ℵ1 [strictly weaker than the Continuum Hypothesis (CH)]. A related assumption strictly weaker than the Generalised CH allows us to prove that ED is fit, that is, contains a dense subspace whose codimension in ED is dim (ED), the largest imaginable. Thus the two components are extreme opposites, and E itself is fit if and only if dim (ED) ≥ dim (Ec), in which case there is a choice of basis for which ED = E. Morover, E is non-flat (if and) only if the codimension of E′ is at least in E*. These results ensure latitude in the search for certain subspaces of E* transverse to E′, as in the barrelled countable enlargement (BCE) problem, and show that every non-flat GM-space has a BCE.