TITLE: Usefulness of Frechet versus Gateaux differentiability or something in between.
QUESTION [13 upvotes]: If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability.  For $x,u \in L$, $V$ is Gateaux differentiable at $x$ in the direction $u$ if the limit $\underset{t \rightarrow 0}{\lim} \frac{V(x + tu) - V(x)}{t}$ exists.  Supposing that you are in a Banach space, $V$ is Frechet differentiable if the above limit exists for all $u$ in a ball around $x$, and importantly, with the convergence being uniform over this neighborhood.  
The question is, what difference does it make for a function to be Frechet differentiable versus Gateaux differentiable, maybe with respect to proving theorems that generalize the finite-dimensional setting, where the two notions of differentiability more or less agree.  What kind of pathological behavior can functions exhibit that are merely Gateaux differentiable in every direction?  There are also intermediate forms of differentiability between Frechet and Gateaux, defined in terms of uniform convergence of the difference quotients over some preferred family of sets (a bornology).  Are there any intermediate kinds of differentiability that are important?

REPLY [3 votes]: I know this question has been inactive for such a long time, but I will try to add something.
Dieudonne in Treatise on analysis defines an "intermediate" concept of differentiability between Fréchet and Gâteaux ones called quasi-differentiability. Using your notation, $V$ is quasi-differentiable at $x$ if $V \circ g$ is differentiable at 0, for all $g : [0,1] \to L$ which are continuous, right-differentiable in $0$, and $g(0) = x$.
(I am not sure about it, but i think you could caracterize quasi-differentiability in terms of uniform convergence on compact sets, i.e. the limit $ \lim_{t\to 0} \frac{V(x+tu) - V(x)}{t}$ ` is uniform on compact sets.)
If $L$ is finite dimensional, then quasi-differentiability implies Fréchet differentiability. If $V$ is Lipschitz, then Gâteaux differentiability implies quasi-differentiability.
Nice thing about quasi-differentiability is that the theorem of differentiation of composite functions holds (wich is not the case with Gâteaux differentiability). Moreover, if $F$ is quasi-differentiable and $G$ is Gâteaux differentiable, then $F \circ G$ is G-differentiable.
Unfortunately, it's possible to construct a function $F : L^2 \to L^2$ which is everywhere quasi-differentiable, its differential is a right inverse (thus injective), but is nowhere locally injective. This is not possible if $F$ is Fréchet differentiable.