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I am having a hard time studying Gateaux derivatives (see https://en.wikipedia.org/wiki/Gateaux_derivative), it seems that every author mentions the concept but only as a cliffhanger to study Fréchet derivatives. My questions are:

1 Is there a good reference on Calculus on Banach\Hilbert spaces that develops the theory using Gateaux derivatives?

2 A good reference on the existence of Taylor remainders theorem for Gateaux differentiable functions?

It is worth mentioning that I have seen many links, but none of them would really treat the subject (see: https://sites.stat.washington.edu/jaw/COURSES/580s/581/LECTNOTES/ch7c.pdf, https://people.tamu.edu/~f-narcowich//m642/m642_notes/F_G_derivatives.pdf, https://www.m-hikari.com/ams/ams-password-2008/ams-password17-20-2008/behmardiAMS17-20-2008.pdf). And that on Wikipedia they mention the Taylors remainder theorem but with no reference attached.

Also this link: Taylor expansion with remainder on locally convex spaces, refers to a paper that may contain the Taylors theorem using Gateaux derivatives but the hyperlink is not working so I can not know if it suits to me or not.

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    $\begingroup$ Unfortunately the situation concerning the book I mentioned has not changed much. I have now a much newer improved version in front of me, but that is useless for you... $\endgroup$
    – Alexander Schmeding
    Commented Jan 15 at 15:59
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    $\begingroup$ I don't know if Gateaux derivatives are really relevant for Question 1. In the context of Banach/Hilbert manifolds, with a calculus based on the Fréchet derivative, quite adequate references are Lang's Differential and Riemannian manifolds and Dieudonné's Foundations of Modern Analysis. $\endgroup$
    – Igor Khavkine
    Commented Jan 16 at 0:25
  • $\begingroup$ Do you have a response to the answers below? $\endgroup$
    – Iosif Pinelis
    Commented Jan 17 at 14:36
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    $\begingroup$ Naive question: What's a situation where a Gateaux derivative is needed but a Fréchet derivative won't do? $\endgroup$
    – Deane Yang
    Commented Jan 27 at 17:26
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    $\begingroup$ @IosifPinelis, not exactly. I wasn’t looking for an example but perhaps a theorem where Gateaux differentiabilit is the “right” one to use. In most cases, Frechet differentiability is needed. $\endgroup$
    – Deane Yang
    Commented Jan 28 at 2:10

2 Answers 2

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(There should be only one question in one post.)

Anyhow, concerning your first question: There is nothing here really to study, as the Gateaux derivative at a given point in a given direction is just the derivative at a given point of a function of a real variable.

Concerning your second question: The Taylor expansion with a remainder is therefore just the Taylor expansion with a remainder of a function of a real variable. This Taylor expansion can be proved in a couple of lines by, say, repeated integration by parts, if the function is smooth enough.

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As was discussed in the comments, my answer on a related question ( Taylor expansion with remainder on locally convex spaces) would be useful if the book was out. Unfortunately not much has changed as the book by Glöckner and Neeb is nearing completion.

The good news is, that I talked since with Neeb and he told me that he would be willing to send the material on request to interested parties as long as the book is not out (which according to him should be in 2024 on the arXiv). So it looks like you can get the Taylor chapter directly by writing an email to Karl-Hermann.

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