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Let's take [itex]E=\mathbb{R}^2, F=\mathbb{R}[/itex]. Consider a function [itex]f:U\subset\mathbb{R}^2\rightarrow\mathbb{R}[/itex], where U is an open subset of [itex]\mathbb{R}^2[/itex]. Then [itex]D^2 f:U\rightarrow L(\mathbb{R}^2;L(\mathbb{R}^2;\mathbb{R}))[/itex].

Now, I read that for [itex]u\in U, v,w\in\mathbb{R}^2[/itex]

*by definition*[itex]D^2 f(u)\cdot (v,w):=D((Df)(.)\cdot w)\cdot v[/itex]. My question now is: Why was this defined precisely this way?

Does it have something to do with "using the product rule", which would amount to [itex]D((Df)(.)\cdot w)=D^2 f(.)\cdot w+Df(.)\cdot D(w)=D^2 f(.)\cdot w[/itex]?

Thanks for any help. Best regards...Cliowa