Fréchet versus Gâteaux differentiability in general topological vector spaces

Determine whether, for single-valued mappings between Hausdorff topological vector spaces equipped with topologies induced by families of F-seminorms, Fréchet differentiability at a point (in the sense of Definition 4.8) implies Gâteaux differentiability at the same point (in the sense of Definition 4.1) without additional assumptions.

Background

In normed vector spaces, Fréchet differentiability is stronger than Gâteaux differentiability, i.e., Fréchet differentiability implies Gâteaux differentiability and the derivatives coincide. The paper proves an implication under extra conditions (Theorem 4.18), but notes that the general case is not established.

This problem asks to bridge that gap by determining whether the implication holds in full generality for topological vector spaces defined via F-seminorms.