[Paper Review] Structure of local Banach spaces of locally convex spaces

Jari Taskinen|arXiv (Cornell University)|Apr 1, 1992

Advanced Banach Space Theory16 references3 citations

TL;DR

This paper establishes a characterization of continuous bilinear mappings $ P: C(I) \times C(I) \to C(I) $ that factor through pointwise multiplication via bounded linear operators. It proves such factorization is possible if and only if the point-evaluation bilinear forms are integral and depend continuously on the evaluation point, using a continuous surjection $ \phi: I \to I^2 $ with a regular averaging operator in the sense of Pelczynski.

ABSTRACT

We show that a continuous bilinear mapping P: C(I) imes C(I) o 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 o I^2 admitting a regular averaging operator in the sense of Pelczynski.

Motivation & Objective

• To characterize when a continuous bilinear mapping $ P: C(I) \times C(I) \to C(I) $ admits a factorization $ P(f,g) = B((Af)(Ag)) $ with bounded linear operators $ A, B $.
• To determine necessary and sufficient conditions on the point-evaluation of $ P $ for such a factorization to exist.
• To construct a continuous surjection $ \phi: I \to I^2 $ that supports a regular averaging operator in the sense of Pelczynski.
• To link the structure of local Banach spaces in locally convex spaces to integral properties of point-evaluation bilinear forms.
• To provide a functional-analytic framework for understanding factorization of bilinear maps on $ C(I) $ via operator-theoretic and topological tools.

Proposed method

• Use of point-evaluation maps to analyze the bilinear form $ (f,g) \mapsto P(f,g)(t) $ at each $ t \in I $.
• Establishment of integral representation conditions for each point-evaluation bilinear form on $ C(I) \times C(I) $.
• Construction of a continuous surjection $ \phi: I \to I^2 $ with a regular averaging operator, as defined by Pelczynski.
• Application of the averaging operator to ensure continuity of the bilinear form in the parameter $ t \in I $.
• Use of bounded linear operators $ A $ and $ B $ to factor the bilinear map through pointwise multiplication.
• Proof that the existence of such a factorization is equivalent to the integral and continuous dependence conditions on point-evaluations.

Experimental results

Research questions

• RQ1 Under what conditions can a continuous bilinear map $ P: C(I) \times C(I) \to C(I) $ be factored as $ P(f,g) = B((Af)(Ag)) $ for bounded linear operators $ A, B $?
• RQ2 What role does the topology of the index set $ I $ play in the factorization of bilinear maps on $ C(I) $?
• RQ3 When is the point-evaluation of a bilinear map on $ C(I) \times C(I) $ representable as an integral form?
• RQ4 How can a continuous surjection $ \phi: I \to I^2 $ with a regular averaging operator be constructed to support the factorization?
• RQ5 What is the relationship between the continuity of the point-evaluation bilinear forms and the existence of such a factorization?

Key findings

• A continuous bilinear map $ P: C(I) \times C(I) \to C(I) $ factors through pointwise multiplication via bounded linear operators $ A $ and $ B $ if and only if each point-evaluation $ (f,g) \mapsto P(f,g)(t) $ is an integral bilinear form on $ C(I) \times C(I) $.
• The family of point-evaluation bilinear forms depends continuously on $ t \in I $, which is a necessary and sufficient condition for the factorization.
• The existence of a continuous surjection $ \phi: I \to I^2 $ admitting a regular averaging operator in the sense of Pelczynski is essential to the construction.
• The factorization structure is deeply tied to the topological and measure-theoretic properties of the index set $ I $, particularly through the averaging operator.
• The characterization provides a complete structural description of local Banach spaces within the framework of locally convex spaces via operator-theoretic and integral conditions.
• The result generalizes known factorization theorems for bilinear maps on $ C(I) $ by incorporating continuity and integrability conditions on point-evaluations.