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Results tagged with big-list
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user 2554
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
40
votes
Probabilistic proofs of analytic facts
Question: Given $n$ points in Euclidean space (which we might as well take to be $\ell_2^n$), what is the smallest $k=k(n)$ so that these points can be moved into $k$-dimensional Euclidean space via …
- 31.5k
37
votes
Proofs of the uncountability of the reals
Alternatively,
Prove that the reals are connected.
Prove that every countable dense subset $X$ of the reals must be order isomorphic to the rationals.
Prove that the rationals are not connected.
- 31.5k
30
votes
How helpful is non-standard analysis?
Nonstandard hulls of spaces are used all the time in Banach space theory, so much so that books devote sections to the construction of ultraproducts of Banach spaces (e.g. Absolutely summing operators …
- 31.5k
29
votes
Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
Tim, here is one very specific example that a computer scientist who cares only about $L_1$ and $L_2$ should find appealing. The norm of $\ell_1^n$ is, up to a constant, the same as the $\ell_p^n$ no …
23
votes
Quick proofs of hard theorems
Lomonosov's 1973 proof that every compact operator $T$ has a hyperinvariant subspace (i.e., a subspace that is invariant for every operator that commutes with $T$) was much simpler than proofs existin …
21
votes
Accepted
A book you would like to write
Gosh, what a question, Gil. What is your answer?
I have written many books in my head, but I am much too lazy actually to write a book. I guess my first choice would be
Geometric nonlinear function …
19
votes
Theorems first published in textbooks?
It happened to me once. While visiting the Institute for Advanced Studies at the Hebrew University of Jerusalem in 1976-77 I answered a question from the preliminary manuscript of volume 1 of Lindens …
17
votes
Are there proofs that you feel you did not "understand" for a long time?
The proof that the trace is well defined for square matrices looked like symbol pushing to me. Many years later I realized that the proof is nonsense if you live in certain infinite dimensional world …
15
votes
Famous mathematical quotes
Jean Bourgain, in response to the question, "Have you ever proved a theorem that you did not know was true until you made a computation?" Answer: "No, but nevertheless it is important to do the comp …
14
votes
What are some reasonable-sounding statements that are independent of ZFC?
My favorite is the first problem I worked on, back in 1966 when I was an undergraduate. The question is: does every non separable Banach space have an uncountable biorthogonal system?
Shelah construc …
12
votes
Your favorite surprising connections in mathematics
I agree with Zavosh that Jones' linking of Von Neumann algebras to knot theory is one of the great connections in modern times. Closer to home for me is Pisier's use of a theorem of Beurling on holom …
10
votes
What are some results in mathematics that have snappy proofs using model theory?
There are many results in Banach space theory that are proved via ultraproducts or non standard hulls, and most books on the subject contain a few. One nice one that is easy to state is that a Banach …
- 31.5k
9
votes
Fixed point theorems
I forgot who proved it, but the statement is nice and very easy to prove: A function $f:X\to X$ is fixed point free if and only if there is a partition of $X$ into three subsets s.t. $f$ maps each of …
7
votes
Mathematicians who were late learners?-list
R. H. Bing taught high school for several years before entering graduate school.
7
votes
Which popular games are the most mathematical?
In bridge, missing QJxx in a suit, if the Q or J drops on the first round, it is better to finesse if possible on the second round if nothing else is known about the distribution. This is obvious to …