User:Boris Tsirelson/Sandbox1: Difference between revisions
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| volume = 85 | | volume = 85 | ||
| pages = 61–78 | | pages = 61–78 | ||
| url = http://www.ams.org/mathscinet | | url = http://www.ams.org/mathscinet-getitem?mr=983381 | ||
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(Also [http://personnel.univ-reunion.fr/ardm/inefff.pdf here].) | (Also [http://personnel.univ-reunion.fr/ardm/inefff.pdf here].) |
Revision as of 14:01, 1 September 2010
The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. If X is similar to a part of Y and at the same time Y is similar to a part of X then X and Y are similar. In order to be specific one should decide
- what kind of mathematical objects are X and Y,
- what is meant by "a part",
- what is meant by "similar".
In the classical Cantor–Bernstein–Schroeder theorem
- X and Y are sets (maybe infinite),
- "a part" is interpreted as a subset,
- "similar" is interpreted as equinumerous.
Not all statements of this form are true. For example, let
- X and Y are triangles,
- "a part" means a triangle inside the given triangle,
- "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").
Then the statement fails badly: every triangle X evidently is similar to some triangle inside Y, and the other way round; however, X and Y need no be similar.
Notes
References
Bourbaki, Nicolas (1968), Elements of mathematics: Theory of sets, Hermann (original), Addison-Wesley (translation).
Casacuberta, C & M Castellet, eds. (1992), Mathematical research today and tomorrow: Viewpoints of seven Fields medalists, Lecture Notes in Mathematics, vol. 1525, Springer-Verlag, ISBN 3-540-56011-4.
Feynman, Richard (1995), The character of physical law (twenty second printing ed.), the MIT press, ISBN 0 262 56003 8.
Gowers, Timothy, ed. (2008), The Princeton companion to mathematics, Princeton University Press, ISBN 978-0-691-11880-2.
Gowers, W.T. (1996), "A solution to the Schroeder-Bernstein problem for Banach spaces", Bull. London Math. Soc. 28: 297–304.
Casazza, P.G. (1989), "The Schroeder-Bernstein property for Banach spaces", Contemp. Math. 85: 61–78. (Also here.)