All Questions
6,260 questions
2
votes
1
answer
927
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Theta Functions and Cousins
So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...
- 27.5k
1
vote
1
answer
322
views
Request for info on the space of commuting matrices preserving a flag.
Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn.
Consider the space of pairs of commuting linear transformations A and B such that:
A preserves the flag (i.e. A(Vi) is in Vi), and
B strictly ...
- 44.7k
43
votes
6
answers
9k
views
"A gentleman never chooses a basis."
Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.
Question. Is there a gentlemanly way to prove that the natural map from $V$ to $V^{**}$ is surjective if $V$...
- 5,275
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
7
votes
2
answers
2k
views
What's the correct notion of determinant of a bilinear pairing?
By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
- 54.6k
18
votes
2
answers
3k
views
Zeta-function regularization of determinants and traces
The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let A be an operator (on an infinite-dimensional ...
- 54.6k
35
votes
5
answers
13k
views
Linearity of the inner product using the parallelogram law
A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula:
$2\langle u,v\rangle ...
- 26.8k
9
votes
6
answers
8k
views
How to approximate a solution to a matrix equation? [closed]
Suppose a matrix equation $Ax = b$ has no solution ($b$ is not in the column space of $A$)
How can I find a vector $x^\prime$ so that $Ax^\prime$ is the closest possible vector to $b$?
- 461
62
votes
9
answers
23k
views
Can a vector space over an infinite field be a finite union of proper subspaces?
Can a (possibly infinite-dimensional) vector space ever be a finite union of proper subspaces?
If the ground field is finite, then any finite-dimensional vector space is finite as a set, so there are ...
- 24.1k
37
votes
4
answers
12k
views
Finite extension of fields with no primitive element
What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
- 24.1k