Uniqueness of the Fréchet derivative without (SC) and (SB) assumptions

Prove the uniqueness of the Fréchet derivative VT(x), as defined in Definition 4.8, for single-valued mappings between Hausdorff topological vector spaces without assuming (SC) that the domain space is seminorm constructed and (SB) that the family of F-seminorms on the domain is bounded in the sense sup{p(u): p in Fx} < ∞ for every u in X.

Background

Theorem 4.10 establishes uniqueness of the Fréchet derivative under two assumptions: (SC) the domain space is seminorm constructed, and (SB) a boundedness condition on the family of F-seminorms. The authors deem these conditions strong and ask whether they can be removed.

A positive resolution would strengthen the theory by guaranteeing uniqueness of the Fréchet derivative in a much broader class of topological vector spaces.