Is there a simple example of an isometry between normed vector spaces that is not an affine map?
Example of a non-linear isometry?
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10How about translation? – 2011-08-24
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3@Q Translations are [affine](http://en.wikipedia.org/wiki/Affine_transformation) by definition. – 2011-08-24
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1@whuber : They are (generally) not linear, which is what the question was asking when QY commented. – 2011-08-24
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0@Ricky Thanks. The translation comment makes perfect sense in light of the original version. – 2011-08-24
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2@Name: This string of comments shows why it's a good idea to mark your edits when you change the content of the question. – 2011-08-24
2 Answers
The note by Jussi Väisälä linked to by the Wikipedia article about the Mazur–Ulam theorem contains the following example:
An isometry need not be affine. To see this, let $E$ be the real line $\mathbf{R}$, let $F$ be the plane with the norm $\lVert x \rVert = \max(|x_1|, |x_2|)$, and let $\phi: R \to R$ be any function such that $|\phi(s)-\phi(t)| \le |s-t|$ for all $s, t \in\mathbf{R}$, for example, $\phi(t) = |t|$ or $\phi(t) = \sin t$. Setting $f(s) = (s, \phi(s))$ we get an isometry $f : E \to F$, which is usually not affine.
(But of course this is not a bijection.)
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1+1 for *linking* to Väisälä's article. For those with the relevant university subscription or near a library may want to look at the [published version](http://www.jstor.org/stable/3647749) in The American Mathematical Monthly Vol. 110, No. 7 (Aug. - Sep., 2003), pp. 633-635. – 2011-08-24
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0@Theo: And thank you too, for linking to the real thing. :) – 2011-08-24
Yes.
Let $\mathbb{C}$ be a vector space over itself with absolute value as its norm.
Define $ \; \; f : \mathbb{C} \to \mathbb{C} \; \; $ by $ \; \; f(z) = \overline{z} \; \; $ .
$f$ is a non-linear (bijective) isometry that satisfies $\; f(0) = 0 \;$ .
See the Mazur–Ulam theorem.
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6This feels a little like cheating, since the corresponding transformation on $\mathbb{R}^2$ *is* linear. ;-) – 2011-08-24
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0@Hans: I agree. The question was posed on normed vector spaces, and as a map on normed vector spaces, $z\mapsto\bar{z}$ is linear. – 2011-08-24
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4No, it's: as a map on _real_ normed vector spaces, $z\mapsto \overline{z}$ is linear. – 2011-08-24
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0@RickyDemer Could you explain how it is non-linear? – 2018-02-12
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2@VivianL. $f(i\hspace{-0.02 in}\cdot \hspace{-0.04 in}1) \: = \: f(i) \: = \: \overline{i} \: = \: -i \: \neq \: i \: = \: i\cdot 1 \: = \: i\cdot \overline{1} \: = \: i\cdot f(1) \hspace{1.54 in}$ – 2018-02-12
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0@RickyDemer When we choose a scalar, we have to make sure it only scales rather than rotate a vector. How can you choose $i$ as a scalar? It probably will rotate a vector. Sorry, I am new to linear algebra. – 2018-02-12
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1@VivianL. I can choose $i$ as a scalar because I specified $\mathbb{C}$ as a vector space $\hspace{1.43 in}$ _over itself_ (rather than over $\mathbb{R}$). – 2018-02-12