I was reading about functional analysis and it interested me greatly applying the techniques of analysis to function spaces but it seems to have abruptly stopped when getting to the derivative. I know the generalized concept of a derivative on a vector space is called a Frechet derivative.

So what would be the Frechet derivative of a functional operator (a function on standard real functions)? For example what is the Frechet derivative of the basic real analysis derivative? I believe it would just be the the basic derivative since that it already a linear operator. Are there any more more interesting Frechet derivatives of other non-linear functional operators? Is there a name for this?

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    $\begingroup$ Frechet derivative of a linear operator is equal to the operator. There is already a name for this: 'Frechet derivative'. Not sure, what you are asking for. $\endgroup$ – daw Oct 23 '18 at 6:16

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