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Results tagged with big-list
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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.
4
votes
Prominent non-mathematical work of mathematicians
Per Enflo is famous for solving Banach's basis problem, Grothendieck's approximation problem, and the invariant subspace problem for general Banach spaces, and has other fundamental research in linear …
- 31.5k
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 …
- 4,713
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 …
- 31.5k
5
votes
Interesting examples of generic behavior of mathematical objects being either unreasonably s...
Their are many such examples in the theory of finite dimensional Banach spaces. Suppose that $X$ is an $n$ dimensional Banach space. If you take a random subspace of dimension $k$, then for some value …
- 31.5k
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 …
- 31.5k
6
votes
Fundamental problems whose solution seems completely out of reach
Is every complemented subspace of $C[0.1]$ isomorphic to $C(K)$ for some compact metric space $K$?
Is every infinite dimensional complemented subspace of $L_1[0.1]$ isomorphic either to $L_1[0.1]$ or …
- 31.5k
6
votes
Accepted
Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?
Say that a Banach space $X$ has property $K$ (for Kummers) provided every subspace of $X$ that is isomorphic to $X$ is complemented. The classical separable, infinite dimensional spaces that have pro …
- 31.5k
0
votes
Awfully sophisticated proof for simple facts
If $0\le f_n \le 1$ is a sequence of continuous functions on $[0,1]$ that converges pointwise to $0$, then $\int_0^1 f_n(t) dt $ converges to $0$. Understandable by freshman, the statement is hard to …
- 31.5k
3
votes
Applications of Brouwer's fixed point theorem
One standard consequence of Brouwer's theorem is Borsuk's antipodal mapping theorem, which in turn is used to prove that if $E$, $F$ are subspaces of a normed space and the dimension of $E$ is strictl …
- 31.5k
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 …
- 31.5k
2
votes
Individual mathematical objects whose study amounts to a (sub)discipline?
$C[0,1]$. Since every separable metric space embeds isometrically into $C[0,1]$ and every separable Banach space embeds isometrically isomorphically into $C[0,1]$, the study of $C[0,1]$ includes the …
- 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
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 …
- 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 …
- 31.5k
7
votes
Mathematicians who were late learners?-list
R. H. Bing taught high school for several years before entering graduate school.
- 31.5k