0
votes
1answer
31 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
1
vote
1answer
52 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
-1
votes
3answers
103 views

Linear algebra. Find a counter-example

this is the statement: if $\vec v_{1}, \vec v_{2} , \vec v_{3}, \vec v_{4}$ is a basis for the vector space $\Bbb R^{4} $, and W is a subspace of $\Bbb R^{4}$, then some subset of the $\vec v$ 's is a ...
0
votes
3answers
121 views

A finite dimensional vector space that is not naturally isomorphic to its dual.

I need an example of a finite dimensional vector space $V$ that is not naturally isomorphic to $V^\ast$. I know that, at least in finite dimensional case, there is a one-to-one correspondence between ...
13
votes
6answers
315 views

Other guises for the vector space $\mathbb{R}^n$?

One way the vector space $\mathbb{R}^n$ can come up is as the space of polynomials over $\mathbb{R}$ of degree at most $(n-1)$ . Here we have the isomorphism: $$(a_0,a_1,\ldots,a_{n-1}) ...
15
votes
4answers
1k views

Is a vector space over a finite field always finite?

Definition of a vector space: Let $V$ be a set and $(\mathbb{K}, +, \cdot)$ a field. $V$ is called a vector space over the field $\mathbb{K}$ if: V1: $(V, +)$ is a commutative group V2: $\forall ...
9
votes
2answers
205 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
0
votes
1answer
260 views

Prove Axiom $10$ (Vector Spaces) independent of the others [duplicate]

Possible Duplicate: Is it possible to construct a quasi-vectorial space without an identity element? In Apostol Multivariable Calculus, $1.5$ exercise $30 b$, he asks the reader to prove ...
2
votes
4answers
433 views

Example for a proper dense subspace?

I have been reading some books on functional analysis, and many of them keep talking about a vector space along with a dense proper subspace of it (especially when constructing counterexamples). But ...