Structure of local Banach spaces of locally convex spaces

TL;DR

This paper characterizes when a bilinear map on continuous functions can be represented via bounded linear operators and pointwise multiplication, using a continuous surjection with a regular averaging operator.

Contribution

It provides a necessary and sufficient condition for representing bilinear maps on C(I) with bounded operators and introduces a construction involving a continuous surjection with averaging properties.

Findings

Characterization of bilinear maps via bounded operators and multiplication.

Construction of a continuous surjection with a regular averaging operator.

Conditions for integral dependence of bilinear forms.

Abstract

We show that a continuous bilinear mapping P: C(I) \times C(I) \to C(I) can be presented in the form P(f,g) = B((Af)(Ag)), where A and B are bounded linear operators on C(I) and multiplication is defined pointwise, if and only if for all t in I the bilinear form (f,g) -> P(f,g)(t) is integral on C(I) times C(I) and depends in a sense continuously on t. To this end we construct a continuous surjection phi : I \to I^2 admitting a regular averaging operator in the sense of Pelczynski.