Definition talk:Differentiable Functional

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I don't know if the book in question gives one, but would you mind figuring out a formal definition? It seems like you'd need to do something like replace the $\epsilon$ by a function and requiring that it tends to zero with $h$, or something. Depends on what $\Delta$ means.

Also: is the convergence $\epsilon\to0$ uniform in $y$? I guess not, because that would be strong. Locally uniform maybe? --barto (talk) 14:53, 30 April 2017 (EDT)

Compare with the definition of differentiablilty in Banach spaces. --barto (talk) 14:56, 30 April 2017 (EDT)

The author of these pages has previously communicated that he is not prepared to put them into any semblance of order or conformity, as he is of the opinion (not shared by all editors) that such things are of negligible importance compared with the actual placing of this material online.
I have the Gelfand and Fomin work on my own shelf, so (once I've finished enjoying myself with these entertaining little morsels of recreational mathematics) I may take some time to go through this series of results myself and do the necessary work, but until then we may find it best to watch this category from a distance, so to speak. --prime mover (talk) 15:09, 30 April 2017 (EDT)
Ok, no problem. --barto (talk) 15:18, 30 April 2017 (EDT)
Correction - not negligible, just not on the top of priority list.
$\Delta J$ is defined here. As for the type of convergence, it is not mentioned. At most I am assuming the limit $\displaystyle \lim_{\size h \mathop \to 0} \epsilon = 0$, where $\size h$ is a norm in function space, and in this case is not defined. The norm itself is described in the book, but for this definition it is not clear which norm he has in mind. Probably the one, which suits the problem in question the best. The author does not consider dependence of $\epsilon$ on $y$ to a great extent. All he wants that the second term would be just slightly steeper than linear in $\size h$.
Indeed it looks like differentiability in Banach spaces, but then it is more of relevance to functional analysis. At this moment I am limited to calculus of variations with respect to variational problems, and I want to improve coverage, and then the style of presentation, hence I would like not to completely change my focus. The hope remains that when I begin comparison with other sources, a better definition will arise. If anyone is planning to work on functional analysis, the contributor's input would be more than welcome. For now I see this category being connected to or spanning articles in classical extremization problems and Lagrangian/Hamiltonian physics. It is too early to couple this to other fields yet, but from my experience functional analysis and theorems in differential equations seem to be demanded the most. Julius (talk) 08:09, 1 May 2017 (EDT)