Differential of function on Hilbert space or Banach space .

Question

In the picture below, I think the life part of 9.1 is something like directional derivative. But in the right part , I don't know what $dE(u)$ is ,I guess it should be a function, but when $E: H\rightarrow R$ , how to define $dE(u)$ ?Is it G derivative or F derivative ?

Answer 1

Whether the author of your book talks about Frechet- or Gateaux-derivatives, is not clear to me from the context, but I'd say the Frechet derivative is more probably. For a function $E \colon H \to \mathbf R$, and a $u \in H$, we say that $E$ is differentiable at $u$, with (Frechet-)derivative $dE(u) \colon H \to \mathbf R$, iff $dE(u) \colon H \to \mathbf R$ is linear and continuous with $$ E(u+h) = E(u) + dE(u)h + o(\|h\|),\qquad h \to 0 $$ Here $dE(u) \colon H \to \mathbf R$ is a linear functional on $H$, and hence can (by Riesz) represented as $\left<\nabla E(u), \cdot\right>$ for some vector $\nabla E(u) \in H$.