On page $106$ in this thesis the differential and gradient are defined in the following manner:
Let $\mathcal{H}$ be a Hilbert space and $f:U\subset \mathcal{H} \rightarrow \mathbb{R}$ for some open set $U$. The differential of $f$ is the element $df \in \mathcal{H}^*$ in the dual space $\mathcal{H}^*$ such that $df(e) = Df\cdot e$. The gradient $\nabla f$ is the map $\nabla f:U\rightarrow \mathcal{H}$ defined by $\langle \nabla f, e \rangle = Df\cdot e$.
My issue with the above is that $df \in \mathcal{H}^*$, while it seems to me that it should read $df : U \rightarrow \mathcal{H}^*$ instead, since the derivative $Df$ is defined earlier (definition $2.56$ page $105$) as $Df : U \rightarrow L(\boldsymbol{E}, \boldsymbol{F})$ (for this instance $L(\boldsymbol{E}, \boldsymbol{F}) = \mathcal{H}^*$, $\boldsymbol{E} = \mathcal{H}, \boldsymbol{F} = \mathbb{R}$). Am I misunderstanding something, or is my interpretation correct?
It seems it is a mistake in the thesis. I am assuming the author either meant to say $df_{x} \in \mathcal{H}^*$ and $df_x(e) = Df_x\cdot e$, or the above interpretation where $df : U \rightarrow \mathcal{H}^*$.