Frechet Derivative
# Definition
Suppose $V, W$ are Normed Vector Spaces and $f: V \to W$. Suppose for $v \in V$ there exists $T \in BL(V, W)$ s.t.
$$\lim\limits_{h \to 0} \frac{|f(v + h) - f(v) - Th|{W}}{|h|{V}} = 0$$
Then we say $f$ has Frechet Derivative $D_{v} f =T$.
# Properties
# Continuity of $f$
# Statement
If $D_{v} f$ exists for $v \in V$, then $f$ is Continuous Function at $v$. TODO
# Proof
Existence of the Frechet Derivative means $$\begin{align*} &&\lim\limits_{h \to 0} \frac{|f(v + h) - f(v) - Th|{W}}{|h|{V}} = 0\\ &\Rightarrow& 0 = \lim\limits_{h \to 0} \Big| \big( f(v + h) - f(v) - Th\big) - \mathbf{0}{W} \Big|{W}\\ \end{align*}$$ By definition of Function Limit, we have that $$\begin{align*} &\lim\limits_{h \to 0} \big( f(v + h) - f(v) - Th\big) = \mathbf{0}{W}\\ &\Rightarrow \lim\limits{h \to 0} \big( f(v + h) - f(v) \big) - \lim\limits_{h \to 0} Th = \mathbf{0}{W}\\ &\Rightarrow \lim\limits{h \to 0} \big( f(v + h) - f(v) \big) = \mathbf{0}_{W} \end{align*}$$ Where the last line follows because Bounded Linear Maps are Continuous (how to prove this?). The last line is the definition of continuity and $f$ is Continuous Function.