On i-operators on real normed spaces

Abstract

We gradually study i-operators on real vector spaces, on real topological vector spaces, and on real normed spaces. Among several things we prove the existence of real topological vector spaces (different from the James’ space) that are free of continuous i-operators. We also prove that every real normed space can be equivalently renormed to be free of norm i-operators. Examples of spaces of continuous functions not admitting norm i-operators and whose unit sphere is free of convex subsets with non-empty interior relative to it are also found. Finally, we also provide some results on a problem posed by Wenzel in [10].