Talk:Contraction mapping
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Untitled edit
In response to the question: "to anon: what's the anomaly?" on my last edit:
In the first section, 'contractions' and 'non-expansive maps' are defined. Contractions are maps that are Lipschitz continuous with Lipschitz constant <1. Non-expansive maps have Lipschitz constant ≤ 1.
In the operator theory section, a 'contraction' is a linear operator of norm ≤ 1. Thus a contraction is a non-expansive linear operator. Thus in operator theory, contraction means something very different than in the context of the first section. For example, the identity map is a contraction operator but not a contraction mapping, following these definitions.
128.135.100.161 01:20, 3 March 2007 (UTC)
- well, if the issue is < 1 and ≤ 1...ok, there is a difference there. Mct mht 18:40, 3 March 2007 (UTC)
- So to say 'as a special case of the above definition' is a bit misleading... I will trust your judgment on the best way to change the wording, but no change at all does not seem very considerate to readers. 128.135.100.161 20:28, 8 March 2007 (UTC)
- yes, you're right. i disagreed with your previous changes because to emphasize this slight difference seemed somewhat misleading as well. but maybe the difference is not so slight. the contraction mapping theorem requires < 1 and would not apply to the operator case. i am going to revert article to your last version. Mct mht 22:12, 8 March 2007 (UTC)
"A contraction mapping has at most one fixed point." edit
"A contraction mapping has at most one fixed point." Isn't this only true for non-expansive maps? Oops, never mind that. Arthena(talk) 21:11, 21 January 2008 (UTC)
- If this is confusing, the "proof" is just writing down mathematically what it means: If f(x)=x [Definition of fixed point x], then d(f(x), f(y)) = d(x, f(y) < d(x, y) which means means that f(y)≠y. Back in words: f(y) is closer to x than y, so f(y) cannot be the same point as y. Worth adding to the main page? — Preceding unsigned comment added by 2001:980:396E:1:D86C:31F5:6801:F457 (talk) 14:08, 6 January 2014 (UTC)
"Firmly non-expansive mapping" equation malformed? edit
Is the first equation in this section broken? It doesn't make sense to me. Martin Packer (talk) 09:04, 23 May 2014 (UTC)