Strong derivative

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Strong derivative
[math]\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)[/math]
For two normed spaces [ilmath](X,\Vert\cdot\Vert_X)[/ilmath] and [ilmath](Y,\Vert\cdot\Vert_Y)[/ilmath]
and a mapping [ilmath]f:U\rightarrow Y[/ilmath] for [ilmath]U[/ilmath] open in [ilmath]X[/ilmath]

[ilmath]df\vert_{x_0}:X\rightarrow Y[/ilmath] a linear map called the
"derivative of [ilmath]f[/ilmath] at [ilmath]x_0[/ilmath]"


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, [ilmath](X,\Vert\cdot\Vert_X)[/ilmath] and [ilmath](Y,\Vert\cdot\Vert_Y)[/ilmath]
  • A mapping, [ilmath]f:U\rightarrow Y[/ilmath] where [ilmath]U[/ilmath] is an open set of [ilmath]X[/ilmath]
  • Some point [ilmath]x_0\in U[/ilmath] (the point we are differentiating at)

Definition 1

If there exists a linear map [ilmath]L_{x_0}:X\rightarrow Y[/ilmath] such that:

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

Definition 2

If there exists a linear map [ilmath]L_{x_0}:X\rightarrow Y[/ilmath] such that:

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

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


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