Questions tagged [vector-spaces]
The vector-spaces tag has no usage guidance.
12
questions
67
votes
39
answers
9k
views
Results true in a dimension and false for higher dimensions
Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.
Here are two examples:
A positive polynomial not reaching its infimum. ...
17
votes
3
answers
861
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
19
votes
2
answers
1k
views
Exponentiation of vector spaces?
This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...
17
votes
6
answers
3k
views
Does the linear automorphism group determine the vector space?
I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...
16
votes
1
answer
814
views
Examples of vector spaces with bases of different cardinalities
In this question Sizes of bases of vector spaces without the axiom of choice it is said that "It is consistent [with ZF] that there are vector spaces that have two bases with completely different ...
13
votes
2
answers
373
views
Subset of vectors whose sum has a large norm
In Rudin - Real & Complex Analysis we have the following
Lemma 6.3. If $z_1, \ldots, z_n \in \mathbb{C}$ then there is a subset $S \subseteq \{1,\ldots,n\}$ for which
$$\left|\sum_{k \in S} z_k\...
4
votes
1
answer
177
views
Is the free modular lattice linear?
Dedekind proved that the free modular lattice on 3 generators is realisable by the intersections and sums of 4-dimensional subspaces in 8-space. Birkhoff showed that the free lattice is infinite if it ...
3
votes
1
answer
133
views
The "semi-symmetric" algebra of a vector space
If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\...
2
votes
0
answers
248
views
Finite-dimensional vector spaces vs. matrices over a semiring
The category $\mathbf{Vect}_k$ of vector spaces over some field $k$ is weakly equivalent to the category $\mathbf{V}_k$ whose objects are finite sets $n$, and whose morphisms $m\to n$ are $(m\times n)$...
2
votes
1
answer
151
views
The minimal volume of the intersection of two $\mathscr{l}_1$-ball in high dimension
We define
$$B_1(r,c) = \{x\in\mathbb{R}^d : \Vert{}x-c\Vert_1 \le r\}$$
Now for arbitrary constant $r \ge s > 0$, given constant $\epsilon \in (r-s,r+s)$, considering $v \in \mathbb{R}^d$ such ...
1
vote
1
answer
394
views
Dimension of a kernel of a linear map
Let $\mathbb{F}$ be a field of characteristic $2$, $n$ be a positive integer and $f_n:\bigoplus\limits_{i=1}^n\mathbb{F}\sigma_{i}\mapsto \bigoplus\limits_{i,j=1,i<j}^n\mathbb{F}\sigma_{i,j}$ be a ...
1
vote
2
answers
446
views
A description of the kernel of projection map from the tensor algebra to the symmetric algebra $T(V)\to S(V)$
Let $V$ be a vector space over some arbitrary field. Let $T(V)$ and $S(V)$ be the tensor and symmetric algebras over $V$. We have the projection map $T(V)\to S(V)$, given by $x_1\otimes\cdots\otimes ...