Linked Questions
11 questions linked to/from Infinite tensor products
19
votes
17
answers
7k
views
Vector spaces without natural bases
Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural".
To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...
30
votes
6
answers
4k
views
Why are finiteness conditions important (and how to recognize them)?
I think everybody here has met lots of finiteness conditions, like those requiring a vector space to be finite dimensional, an abelian group to be finitely generated, a ring to be Noetherian, a ...
11
votes
3
answers
943
views
$\infty$-ary tensor product on a category
Is there any notion in the literature which captures the idea of an $\infty$-ary tensor product on a category $C$? This should include tensor products of $\alpha$-indexed families of objects in $C$ ...
- 63.1k
15
votes
5
answers
1k
views
Monoids with infinite products
Say a monoid $M$ has infinite products if, for any (possibly infinite) sequence $(m_1,m_2,\ldots)$ of elements of $M$, there exists an element $m_1m_2\cdots\in M$, satisfying some good properties. ...
- 8,659
14
votes
1
answer
2k
views
Infinite tensor product of states
Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher.
Even in the simplest ...
- 143
1
vote
3
answers
812
views
Limit for divergent sequences
Sorry for the title, but I think it's funny. Can you write down a homomorphism (of additive groups)
$\mathbb{R}^\mathbb{N} \to \mathbb{R}$,
which is nontrivial and whose kernel contains the finite ...
- 63.1k
5
votes
1
answer
309
views
(Z/n)^(I) is a direct summand of (Z/n)^I
Dear group theorists,
Let $n \geq 1$ and $I$ be an infinite set (you may assume $I$ to be countable). Is the abelian group $(\mathbb{Z}/n)^{(I)}$ (direct sum of copies $\mathbb{Z}/n$) a direct ...
- 63.1k
3
votes
1
answer
848
views
cardinality of product modulo direct sum
let $(X_i)_{i \in I}$ be an infinite family of sets with $|X_i| \geq 2$. we define an equivalence relation on $X = \prod_{i \in I} X_i$ by $x \sim y \Leftrightarrow \{i : x_i \neq y_i\}$ is finite. ...
- 63.1k
6
votes
1
answer
643
views
Generalization of analytic functors
It's been a long time since I posted the following question on stackexchange.
Now I think it's better to adress it to you, in the hope I will reach the right audience: Martin's comment, albeit useful,...
- 13.6k
6
votes
1
answer
563
views
Split powers of the multiplicative group of a field
Let $K$ be a field, $K^\times$ its multiplicative group and $I$ an infinite set. Is then $(K^\times)^{(I)} \subseteq (K^\times)^I$ a direct summand? If not, is it possible to characterize the fields ...
- 63.1k
3
votes
2
answers
436
views
Is certain topology-related set a distributive lattice?
In my research the following problem appeared (and if it is true, this solves positively several my conjectures):
Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ($n$ ...
- 765