Derivative (analysis)

Definition

There are 2 kinds of derivative, a strong derivative (AKA the Frécet derivative, total derivative) one and a weak derivative one (AKA the Gateaux derivative, directional derivative)

Strong derivative

Strong derivative/Definitions overview

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TODO: Weak derivative

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OLD PAGE

Note to self: don't forget to mention the $h$ or $x-x_0$ thing doesn't matter

Definition

Note: there are 2 definitions of differentiability, I will state them both here, then prove them equivalent.

Let $U$ be an open set of a Banach space $X$, let $Y$ be another Banach space.

• Let $f:X\rightarrow Y$ be a given map
• Let $x_0\in X$ be a point.

Definition 1

We say that $f$ is differentiable at a point $x_0\in X$ if:

• there exists a continuous linear map, $L_{x_0}\in L(X,Y)$ such that:
  • $T(x_0+h)-T(x_0)=L_{x_0}(h)+r(x_0,h)$ where $$\ \lim_{h\rightarrow 0}\left(\frac{\Vert r(x_0,h)\Vert}{\Vert h\Vert}\right)=0$$

Definition 2

We say that $f$ is differentiable at a point $x_0\in X$ if:

• there exists a continuous linear map, $L_{x_0}\in L(X,Y)$ such that:
  • $$\lim_{h\rightarrow 0}\left(\frac{f(x_0+h)-f(x_0)-L_{x_0}(h)}{\Vert h\Vert}\right)=0$$

Hybrid definition

These naturally lead to: We say that $f$ is differentiable at a point $x_0\in X$ if:

• there exists a continuous linear map, $L_{x_0}\in L(X,Y)$ such that:
  • $$\lim_{h\rightarrow 0}\left(\frac{\Vert f(x_0+h)-f(x_0)-L_{x_0}(h)\Vert}{\Vert h\Vert}\right)=0$$

Extra workings for proof

• there exists a continuous linear map, $L_{x_0}\in L(X,Y)$ such that:
  • $T(x_0+h)-T(x_0)=L_{x_0}(h)+r(x_0,h)$ where $$\ \lim_{h\rightarrow 0}\left(\frac{\Vert r(x_0,h)\Vert}{\Vert h\Vert}\right)=0$$
  • This can be interpreted as $$\ \exists L_{x_0}\in L(X,Y)\left[\lim_{h\rightarrow 0}\left(\frac{\Vert r(x_0,h)\Vert}{\Vert h\Vert}\right)=0\implies T(x_0+h)-T(x_0)=L_{x_0}(h)+r(x_0,h)\right]$$
    • Which is
      1. $$\exists L_{x_0}\in L(X,Y)\forall\epsilon>0\exists\delta>0\forall h\in X\left[0<\Vert h-x\Vert<\delta\implies\frac{\Vert r(x_0,h)\Vert}{\Vert h\Vert}<\epsilon\implies T(x_0+h)-T(x_0)=L_{x_0}(h)+r(x_0,h)\right]$$
         • Does this make sense though? We need $r(x_0,\cdot)$ to be given, surely a form with $r(x_0,h)=T(x_0+h)-T(x_0)-L_{x_0}(h)$ in the numerator would make more sense? No of course not.

this sort of outlines the proof I'll need for definitions 1 and 2