Questions tagged [vector-spaces]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
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 ...
Nick S's user avatar
  • 1,990
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 ...
მამუკა ჯიბლაძე's user avatar
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: ...
LSpice's user avatar
  • 11.3k
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 ...
H.D. Kirchmann's user avatar
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\...
user avatar
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 ...
grok's user avatar
  • 2,489
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\...
Constantin-Nicolae Beli's user avatar
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 $(m\times n)$...
David Spivak's user avatar
  • 8,549
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 ...
nowhere's user avatar
  • 21
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 ...
Marcos's user avatar
  • 577
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 ...
Constantin-Nicolae Beli's user avatar