Chain rule for Fréchet differentiability in general topological vector spaces

Establish a chain rule for the Fréchet differentiability defined in Definition 4.8 on Hausdorff topological vector spaces whose topologies are induced by families of F-seminorms: given single-valued mappings T: X -> Y and S: Y -> Z between such spaces, determine conditions under which T is Fréchet differentiable at x in X and S is Fréchet differentiable at T(x) in Y imply that the composition S∘T is Fréchet differentiable at x and that the derivative satisfies V(S∘T)(x) = VS(T(x)) ∘ VT(x).

Background

In Banach spaces, the classical chain rule holds for Fréchet derivatives; see Theorem 2.1 in reference [1]. This paper defines a generalized Fréchet differentiability (Definition 4.8) using families of F-seminorms to handle general topological vector spaces without norms.

The authors explicitly raise whether an analogous chain rule can be established in this broader setting, indicating it as an open direction and referring to the conclusion for further details.

References

Question 4.17. Recall that the Fréchet differentiability of mappings in Banach spaces satisfies the chain-rules (see Theorem 2.1 in [1]). Under the Definition 4.8, does the Fréchet differentiability of mappings in general topological vector spaces also satisfies a certain type of chain-rules? We present this question in details in the conclusion section.

— Differentiation in Topological Vector Spaces  (2603.29170 - Li, 31 Mar 2026) in Question 4.17, Section 4.2