Does this kind of space have a name?

[Someone2841]

Consider the set $V = \left \{ f : f \text{is any real-valued function of one real variable}\right \}$. I believe that $V$ is a vector space over the field $\mathbb{R}$, since for all $f,g \in V$ and $a,b \in \mathbb{R}$, it is true that $0 \in V$, $af-bg \in V$, $af+ag = a(f+g)$, and $(ab)f = a(bf)$. (Forgive me if I've missed something or used strange vocabulary or symbology!)

The thing is, $V$ is not only a group under addition, but also a group under multiplication: $\frac{af}{bg} \in V$. Is there a special name for a vector space that is also a field? Thanks!

 

[andrewkirk]

What is the inverse in the multiplicative group of the function f(x)=x?

 

[Someone2841]

Great point. That totally invalidates my example.

Are there vector spaces that are also fields?

 

[andrewkirk]

I'm pretty sure that only vector spaces of dimension 0 or 1 can be fields. So we have the vector spaces {0} (trivial), $\mathbb{R},\ \mathbb{C},\ \mathbb{Z}_n$ for $z\in\mathbb{N}$.

I think there won't be any vector spaces of dimension 2 or greater that are also fields because one can use the property of closure under addition and scalar multiplication to generate an element that does not have a multiplicative inverse. One can't do that in a one-dimensional field because only the zero element doesn't have a multiplicative inverse, and a special exception is made for that in the field axioms.

 

[Someone2841]

What about the vector space $\mathbb{C}=\{(a,b):a,b \in \mathbb{R} \}$, with addition defined as $(a,b) + (c,d) = (a+c,b+d)$, multiplication defined as $(a,b)(c,d)=(ac-bd,ad+bc)$, $\vec 0 = (0,0)$, $\vec 1 = (1,0)$, and inverses defined by $-(a,b)=(-a,-b)$ (additive) and $(a,b)^{-1}=(\frac{a}{a^2+b^2},-\frac{b}{a^2+b^2})$ (multiplicative)? It may then be scaled by the field $\mathbb{R}$: with $r,s \in \mathbb{R}$, $rs(a,b) = r(sa,sb)$.

Is this any example of a two dimensional vector space that is also a field?

 

[andrewkirk]

By Jove, I think you're right. Good idea!
So scrap my notion about the limit of 1D.
I see from wikipedia that the quaternions are a 4D vector space over the reals that is a division algebra - ie it has every property of a field except commutativity of multiplication.

 

[pasmith]

Consider the set $V = \left \{ f : f \text{is any real-valued function of one real variable}\right \}$. I believe that $V$ is a vector space over the field $\mathbb{R}$, since for all $f,g \in V$ and $a,b \in \mathbb{R}$, it is true that $0 \in V$, $af-bg \in V$, $af+ag = a(f+g)$, and $(ab)f = a(bf)$. (Forgive me if I've missed something or used strange vocabulary or symbology!)

The thing is, $V$ is not only a group under addition, but also a group under multiplication: $\frac{af}{bg} \in V$. Is there a special name for a vector space that is also a field? Thanks!

$V$ is not a field: there are functions other than the additive identity (the constant zero function) which do not have multiplicative inverses, namely any non-constant function which takes the value zero somewhere.

$V$ is, however, a ring under pointwise addition and multiplication.

 

[HallsofIvy]

A field that also has scalar multiplication defined and is a vector space is called an "algebra". An example is the set of all polynomials in x. That is an infinite dimensional vector space with addition and multiplication by numbers. In fact that example is a "graded algebra" since the subset of all polynomials of degree less than or equal to n is a vector space for all n.