Given an Hilbert space $H$ P.L.Lions in [Lions, Pierre-Louis. "Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolutions." Acta mathematica 161 (1988): 243-278.] defines the space $X=\left\{\varphi \in C^1(H ; \mathbf{R}) ; D \varphi\right.$ is Lipschitz on bounded sets of $H$ and for all $h, k \in H$ the limit $$\lim _{t \rightarrow 0}(1 / t)(D \varphi(x+t k)-D \varphi(x), h)$$ exists and is uniformly continuous on bounded sets of $H\}$. Then he claims "By elementary differential calculus considerations, one checks easily that if $\varphi \in X$ then we have $$ \begin{aligned} \lim _{t \rightarrow 0} \frac{1}{t}(D \varphi(x+t k)-D \varphi(x), h) &=(A(x) h, k), \quad \forall x, h, k \in H \end{aligned} $$ where $A(x)$ is a bouded symmetric bilinear form, $\|A(x)\|$ is bounded by the Lipschitz constant of $D \varphi$ on balls of $H$ and $A\left(x_n\right)\to A(x)$ pointwise if $x_n \to x$ in $H$. Furthermore, the limits above are uniform on bounded sets of $H$."
My question is: $A(x)$ is not the Gateaux second order differential of $\varphi$, right? Not Frechet of course as the limits are not uniform on $k$ but not even Gateaux as it is not the Gateaux (first order) differential of $D \varphi$, right?
First of all, at every point $x \in H$, the differential is an element in $H^*$, the dual space. But by the Riesz isomorphism, you can easily identify this as a vector, written as $D\phi(x)$.
Now you consider the mapping $$ D\phi:H \to H, x \mapsto D\phi(x), $$ which is also smooth by assumption. Now, for arbitrary $h \in H$, you show that $$ \lim_{t \to 0}(\frac{1}{t}[D\phi(x+tk)-D\phi(x)],h) \to (A(x)k,h). $$ This is just saying that the "sequence" $$ \lim_{t \to 0}\frac{1}{t}[D\phi(x+tk)-D\phi(x)] \rightharpoonup A(x)k $$ converges weakly to $A(x)k$ in $H$.
So you are just asking for $D\phi$ to admit some "weak Gateaux differential", which can be represented by a linear operator with certain properties. Since weak and strong convergence are not equivalent without further ado, this is not the Gateaux differential.
For the Gateaux differential, you are usually asking for strong convergence of the difference quotient, which you dont here.