Definition 27.2.5.label Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma^{E}_{\text{Fin}}, \sigma^{E}_{c}, \sigma^{E}_{b} \subset 2^{E}$ be the collection of all finite, precompact, and bounded subsets, respectively, then differentiability with respect to $\sigma^{E}_{\text{Fin}}, \sigma^{E}_{c}, \sigma^{E}_{b}$ correspond to Gateaux, Hadamard, and Fréchet differentiability.