Nonstandard Integration Theory in Topological Vector Lattices.
For each natural number N, we give an example of a Banach space X such that the set of norm attaining N-linear forms is dense in the space of all continuous N-linear forms on X, but there are continuous (N+1)-linear forms on X which cannot be approximated by norm attaining (N+1)-linear forms. Actually,X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous result for homogeneous polynomials.
In this paper we present a general “gliding hump” condition that implies the barrelledness of a normed vector space. Several examples of subspaces of are shown to be barrelled using the theorem. The barrelledness of the space of Pettis integrable functions is also implied by the theorem (this was first shown in [3]).
Some basic properties of generalized normed spaces are investigated.
We introduce the spaces , , and of multifunctions. We prove that the spaces and are complete. Also, we get some convergence theorems.