[FOM] Q and A (nonstandard analysis)

[Martin Davis]

On November 6 Gabriel Stolzenberg wrote:
 >Question.  What is the name of the famous conjecture in analysis
 >whose proof by Abraham Robinson is sometimes offered as a demonstration
 >of the power of nonstandard analysis?  (Jim Holt did this in the NYR
 >but, as I recall, without identifying the conjecture.)

The reference is likely to the Bernstein-Robinson theorem:
Let T be a linear operator on Hilbert space H such that for some polynomial p,
p(T) is compact. Then H has a non-trivial closed linear subspace E 
such that T maps E
into itself.

This answered a problem of Paul Halmos that had been open for a long 
time. The proof used non-standard methods in a particularly beautiful 
way, approximating an infinite dimensional space from above by a 
space with hyperfinite dimension, so the theorems of finite 
dimensional linear algebra could be brought to bear.
Soon afterwards generalizations of the theorem were proved by standard methods.

Martin



                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com