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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 11th 2011
   • (edited Nov 11th 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexAfter contributing to the article on [[parallelogram identity]], I added to [[isometry]] and created [[Mazur-Ulam theorem]]. The easy proof added at [[isometry]], that shows an isometry $E \to F$ between normed vector spaces is affine if $F$ is strictly convex, might lead one to suspect that the proof under [[parallelogram identity]] was overkill, but I think that's an illusion. Ultimately, I believe the [[parallelogram identity]] is secretly an expression of the perfect 'roundness' of spheres, connected with the fact observed by Tom Leinster recently at the Café that the group of isometries for the $l_2$ norm is a continuum, whereas for other $p$ in the range $1 \lt p \lt \infty$, you get just a finite reflection group (this is for the finite-dimensional case, but there's an analogue in the infinite-dimensional case as well). The Mazur-Ulam theorem removes the strict convexity hypothesis, but adds the hypothesis that the isometry is surjective. The conclusion is generally false if this hypothesis is omitted.

   After contributing to the article on parallelogram identity, I added to isometry and created Mazur-Ulam theorem. The easy proof added at isometry, that shows an isometry $E \to F$ between normed vector spaces is affine if $F$ is strictly convex, might lead one to suspect that the proof under parallelogram identity was overkill, but I think that’s an illusion. Ultimately, I believe the parallelogram identity is secretly an expression of the perfect ’roundness’ of spheres, connected with the fact observed by Tom Leinster recently at the Café that the group of isometries for the $l_2$ norm is a continuum, whereas for other $p$ in the range $1 \lt p \lt \infty$, you get just a finite reflection group (this is for the finite-dimensional case, but there’s an analogue in the infinite-dimensional case as well).

   The Mazur-Ulam theorem removes the strict convexity hypothesis, but adds the hypothesis that the isometry is surjective. The conclusion is generally false if this hypothesis is omitted.