Frechet Derivative

Last updated Nov 1, 2022

# Definition

s 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 $f$

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.

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Frechet Derivative

# Definition

Normed Vector Space