Talk:Kantorovich theorem

Euclidean space

Why is everything couched in Euclidean space? Is there not a version that applies to any Banach spaces? 203.167.251.186 (talk) 01:22, 15 July 2010 (UTC)

Actually, it is the same. One would have to add technicalities like that source and target space could be different Banach spaces, however the derivative in x0 has to define an homeomorphism of both spaces, one could discuss the choice of different equivalent norms etc. The german version has a paragraph to that effect.--LutzL (talk) 08:29, 15 July 2010 (UTC)

Locally Lipschitz

I think the definition given of locally Lipschitz is wrong. Take the open set to equal to the whole space. Really given any point there should exist an open set, is that correct? — Preceding unsigned comment added by 2607:EA00:107:3C01:6045:2882:4736:C33 (talk) 17:50, 20 November 2015 (UTC)

Definition currently given is:

Let $X\subset \mathbb {R} ^{n}$ be an open subset and $F:\mathbb {R} ^{n}\supset X\to \mathbb {R} ^{n}$ a differentiable function with a Jacobian $F^{\prime }(x)$ that is locally Lipschitz continuous (for instance if it is twice differentiable). That is, it is assumed that for any open subset $U\subset X$ there exists a constant $L>0$ such that for any $\mathbf {x} ,\mathbf {y} \in U$

$\|F'(\mathbf {x} )-F'(\mathbf {y} )\|\leq L\;\|\mathbf {x} -\mathbf {y} \|$

holds. — Preceding unsigned comment added by 2607:EA00:107:3C01:6045:2882:4736:C33 (talk) 17:52, 20 November 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Kantorovich theorem/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This is a support article for Newton's method. One could add a sketch of a proof as done in the german article, but then proofs are not the main topic of an encyclopedia. An image as given in the source would be wellcome.--LutzL 06:13, 14 June 2007 (UTC)

Last edited at 06:13, 14 June 2007 (UTC). Substituted at 02:16, 5 May 2016 (UTC)