Strong derivative

Strong derivative

$$\lim_{h\rightarrow 0}\left(\frac{\big\Vert f(x_0+h)-f(x_0)-df\vert_{x_0}\big\Vert_Y}{\Vert h\Vert_X}\right)$$ For two normed spaces $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ and a mapping $f:U\rightarrow Y$ for $U$ open in $X$ $df\vert_{x_0}:X\rightarrow Y$ a linear map called the "derivative of $f$ at $x_0$"

Definition

The strong derivative (AKA the Fréchet derivative) has several definitions, however they are all equivalent, as will be shown. In all cases we are given:

• Two normed vector spaces, $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$
• A mapping, $f:U\rightarrow Y$ where $U$ is an open set of $X$
• Some point $x_0\in U$ (the point we are differentiating at)

Definition 1

If there exists a linear map $L_{x_0}:X\rightarrow Y$ such that:

• $$f(x+h)-f(x)=L_{x_0}(h)+r(x_0;h)$$ where $$\lim_{h\rightarrow 0}\left(\frac{\Vert r(x_0;h)\Vert_Y}{\Vert h\Vert_X}\right)=0$$

Definition 2

If there exists a linear map $L_{x_0}:X\rightarrow Y$ such that:

• $$\lim_{h\rightarrow 0}\left(\frac{f(x_0+h)-f(x_0)-L_{x_0}(h)}{\Vert h\Vert_X}\right)=0_Y$$

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TODO: Check this, I've just been sick, so I'm going to save my work and lie down

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Todo

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TODO: Find reference for and add "total derivative" to list of AKA names, see also derivative (analysis) and do the same thing there

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