Non-Schwartzian Power Series Spaces.
The aim of this paper, is to introduce the convex structure (specially, Takahashi convex structure) on modular spaces. Moreover, we are interested in proving some common fixed point theorems for non-self mappings in modular space.
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.