Tagged Questions
13
votes
4answers
594 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
174 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
168 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
3answers
224 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 ...
