In their book The Ricci Flow in Riemannian Geometry Andrews and Hopper have an appendix on Gâteaux and Fréchet differentiability.
They define Gâteaux differentiability as follows:
Let $X,Y$ be Banach spaces. A function $f: X \rightarrow Y$ is said to be Gâteaux differentiable at $x$ if there exists a bounded linear operator $T_x \in L(X,Y)$ such that for all $v \in X$, $$\lim_{t \to 0} \frac{f(x+tv)-f(x)}{t} = T_x v.$$ The operator $T_x$ is called the Gâteaux derivative.
This is the definition I was presented with in class. Fréchet differentiability, however, is defined differently than in class (In class we used the second definition mentioned in the following passage):
If the limit (in the sense of the Gâteaux derivative) exists uniformly in $v$ on the unit sphere of X, we say $f$ is Fréchet differentiable at $x$ and $T_x$ is the Fréchet derivative of $f$ at $x$. Equivalently, if we set $y=tv$ then $t \rightarrow 0$ if and only if $y \rightarrow 0$. Thus $f$ is Fréchet differentiable at $x$ if for all $y$, $$f(x+y)-f(x)-T_x(y)=o(\|y\|).$$
Questions
- What does "the limit exists uniformly in $v$ on the unit sphere of X" mean exactly? I know what uniform convergence is in regards to a sequence of functions. However, I cannot figure out how this concept relates to the one presented here, i.e. what sequence of functions is considered here?
- How are the two presented definitions of Fréchet differentiability equivalent?