Talk:Fréchet derivative

message: "Thus, in order for a linear Gâteaux derivative to imply the existence of the Fréchet derivative, the difference quotients have to converge uniformly for all directions"
Converge uniformly?  Where is the sequence of functions?
Should this perhaps read "must be uniformly continuous"?
Or are you saying that the difference quotients should converge to the same value?  — Preceding unsigned comment added by 138.115.172.134 (talk) 18:47, 22 May 2018 (UTC)
replies: text: Hello?  — Preceding unsigned comment added by 138.115.172.134 (talk) 18:27, 15 June 2018 (UTC)
thread_title:  Incredibly confusing sentence 

message: Hi Linas.  I was hoping to get more comfortable with the idea of the Frechet derivative being called holomorphic, and you have a section here which purports to explain it.  But as of now, I can't understand it.  For starters, your notation seems off.  x and y are members of different spaces, but you're adding them.  By what right can you do that?
:Have U\subset V as in previous para, so x+y \in V. Will try to clarify.
What are the domain and codomain of f? 
:Domain is U (hhmm, ok, I guess domain has to be V, see below) and codomain is W, as per first paragraph. 
::Ohh, I see the problem. Will need to ponder, as this is an important distinction, it would seem. I think the intent of what I was reading was that f is defined on V but might only be differentiable on U. Unfortunately, this distinction is, ahem, blurred. ... linas 18:49, 25 September 2005 (UTC)
replies: text: Right, sorry, meant to say L is elt of W*. Will clean up.
I guess the basic idea is clear though: if all linear functionals acting on the function yield holomorphic maps, then the function itself is called holomorphic, right?  -Lethe | Talk 10:05, 25 September 2005 (UTC)

text: Yes, and I assumed this to be the historical origin of the term, from what I can tell. However, its not entirely clear to me just how strong the relationship between h(z) being holomorphic and f being analytic. That is, if h(z) is holomorphic for all functionals L, does this imply that higher derivatives (in fact, all derivatives) of f exist?  This would make a good homework problem. One would need to work through radius-of-convergence arguments, as a minimum. linas 18:28, 25 September 2005 (UTC)
replies: text: Err, I mean, that is essentially the content of the "Taylor's theorem" section. The homework problem is then to really go through the proof at the next level of detail, and convince oneself that it all really does work. I haven't gone through that level of detail; I'm taking this on faith. Thus, while it would be great to add a sentence that stated if h(z) is holomorphic for all functionals L, this implies that higher derivatives (in fact, all derivatives) of f exist to the article, I have not yet worked through this enough to be confident to make this statement. I suspect, though, that this is probably a "well-known" result of the theory. linas 18:39, 25 September 2005 (UTC)

text: I am also skeptical.  I'll try to think of a counterexample.  Best, Silly rabbit 21:50, 16 November 2005 (UTC)
thread_title:  holomorphic 

message: According to the article functional derivative, Fréchet derivatives are defined on Fréchet space.  I suppose we could use the translation invariant metric to define a derivative.  Would that qualify as a Fréchet derivative?  -lethe talk [ +] 01:36, 2 February 2006 (UTC)
replies: text: I've opened an article on Differentiation in Fréchet spaces.  The metric doesn't come into play much, since a metric isn't really part of the data specifying a Fréchet space (which only needs to be metrizable -- an important distinction). Actually, the derivative resembles the Gâteaux derivative more than the Fréchet derivative.  This is probably the weakest notion of differentiation available which satisfies many of the familiar properties from calculus.  Silly rabbit 18:07, 8 June 2006 (UTC)
thread_title:  Fréchet space 

message: What do you think of my section User:Saippuakauppias/Fréchet (in fact, I saw this article much after having written my article. However, I think, we could copy the explication part into this article ("Idea"). What do you think of? - Please post your comments to the articles Talk site. Thanks for your comments! --Saippuakauppias ⇄ 19:27, 28 April 2008 (UTC)
replies: text: Maybe one could add half a sentence under "see also", saying that \langle \mathbf{\nabla f(\mathbf{a})}, \cdot \rangle is the Frechet derivative in case V=R^n and W=R. .. Hm, think about it, maybe it would really help someone who knows the gradient if he understood the difference. ... But I don't think it really gives a complete section, but rather something like two sentences. -- JanCK (talk) 07:52, 29 April 2008 (UTC)
replies: text:  upps, reading the article I see that Finite dimensions already contains this example. It just lags to mention that for m=1 the vector of these  partial derivates is called the gradient. -- JanCK (talk) 07:59, 29 April 2008 (UTC)
thread_title:  New section with the idea? 

message: This page contradicts the article Gateaux derivative. That page says the G-derivative has to be linear, whereas this one doesn't. The difference is that the Gateaux derivative is only a directional derivative. I am quite certain this is the correct definition of the Gateaux. See the discussion  here for more. I think we need to edit this page to sort this out. Here is a quote from that thread:
:A standard example of a function on \R^2 that is differentiable in the origin (0,0) in Gâteaux but not in Fréchet sense, is \textstyle \big(\frac{x^2y}{x^4+y^2}\big)^2, which is not even continuous.
I've put a watch on this article, and I'll be back to try and sort this out by discussion, but I think you'll find, nearly everywhere you look, the Gateaux is defined the same way. It's been emotional (talk) 03:07, 29 April 2009 (UTC)
replies: text: That page doesn't say that the Gateaux derivative must be linear.  In fact, it says at several points that the derivative may fail to be linear.  A standard reference is the treatise by Hille and Phillips.  Sławomir Biały (talk) 03:08, 1 May 2009 (UTC)

text: Ah... that article makes a strange distinction between the differential and the derivative which I hadn't noticed at first.  I think the template should go there, not here.  At least the usage in this article is clear from the context.  Sławomir Biały (talk) 03:16, 1 May 2009 (UTC)

text: I have more or less fixed the problem with the Gâteaux derivative article.  Sławomir Biały (talk) 11:58, 1 May 2009 (UTC)

text: I have brought the article on the Gateaux derivative into accord with the sources there.  As I'm sure you know, reference desk discussions are not reliable sources for use in encyclopedia articles.  The sources are to Max Zorn, Einar Hille, and Ralph Phillips, as well as to the Springer EOM entry.  Now, the Gateaux derivative article is correct to point out that in many cases the derivative/differential is assumed to be linear as part of the definition.  However, this is certainly not always true, even in top-notch sources, as you seem to believe.  At any rate, there is now no further internal contradiction in our Wikipedia articles: the Gateaux derivative article has (since your post) had the usage there cleaned up.  If you still find a contradiction, then please state clearly where the contradiction is.  Sławomir Biały (talk) 15:12, 1 May 2009 (UTC)
thread_title:  contradiction with page on gateaux derivative 

message: :The article states that "if r and s are scalars (two real or complex numbers)..."  This isn't strictly true since we've never asserted that the Banach space W is defined over \mathbb{R} or \mathbb{C}.  —Preceding unsigned comment added by 198.206.219.39 (talk) 15:17, 29 July 2010 (UTC)
replies: text: The purpose of the ambiguity seems to be to allow the Banach spaces to be defined over either R or C.    The field plays pretty much no role at all otherwise, so it's probably not worth making more out of this than necessary.  Sławomir Biały (talk) 15:30, 29 July 2010 (UTC)
thread_title:  Description of linearity 

message: On the real line, they are equivalent, no?  I think this special case should be mentioned -- it also explains why there is no fuss made over the distinction when people first learn single variable calculus. Lavaka (talk) 16:41, 17 March 2011 (UTC)
thread_title:  more on Gâteaux versus Fréchet equivelence 

message: The article states that "A function f that is Fréchet differentiable for any point of U, and whose derivative Df(x) is continuous in x on U, is said to be C1."
But this is trivially true, since all bounded linear operators are continuous. I believe the correct definition should be that f is C1 iff the function that maps every x to Df(x) is continuous.
Cheers  — Preceding unsigned comment added by 84.192.117.114 (talk) 18:27, 18 August 2012 (UTC)
thread_title:  condition for C1 trivial