In set theory, when we talk about the cardinality of a set we have notions of finite, countable and uncountably infinite sets.
Main Question
Let's talk about the dimension of a vector space. In linear algebra I have heard that vector spaces are either of finite dimension (for example $\mathbb{R}^n$) or infinite dimension (for example $C[0,1]$).
Why don't we have notions of countably infinite dimensional and uncountably infinite dimensional vector spaces?
Maybe, I am missing the bigger picture.
Extras
P.S. A long time ago, I attended a talk given on enumerative algebraic geometry and the professor said, "I always think of a positive natural number as the dimension of some vector space".
Can this idea then be extended to vector spaces of uncountably infinite dimension by considering transfinite numbers as denoting the dimension of some vector space?
Best Answer
We do have the notions of countable/uncountable dimensions. Just as a set can be finite or infinite (without specifying which infinite cardinality the set as) a vector space can be finite dimensional or infinite dimensional. We can then go one step more and ask, if the dimension is infinite, which infinite cardinal is it?
The definition of dimension of a vector space is the cardinality of a basis for that vector space (it does not matter which basis you take, because they all have the same cardinality). Then for any cardinal number $\gamma$, you can have a vector space with that dimension. For example, if $\Gamma$ is a set with cardinality $\gamma$, let $c_{00}(\Gamma)$ be the space of all $\mathbb{F}$-valued functions $f$ such that $$\text{supp}(f)=\{x\in \Gamma: f(x)\neq 0\}$$ is finite. Then let $\delta_x\in c_{00}(\Gamma)$ be the function such that $\delta_x(y)=0$ if $y\neq x$ and $\delta_x(y)=1$ if $y=x$. Then $(\delta_x)_{x\in \Gamma}$ is a basis for $c_{00}(\Gamma)$ with cardinality $\gamma$. If $\Gamma=\mathbb{N}$, we have a vector space with countably infinite dimension. If $\Gamma=\mathbb{R}$, we have a vector space with dimension equal to the cardinality of the continuum.
However, for infinite dimensional topological vector spaces (and for infinite dimensional Hilbert and Banach spaces in particular) the usual notion of a basis of limited use. This is because the coordinate functionals for an infinite basis do not interact very well with the topology (one can show that if $(e_i, e^*_i)_{i\in I}$ is a basis together with its coordinate functionals for an infinite dimensional Banach space, then only finitely many of the functionals $e^*_i$ can be continuous). Since the notion of a basis is not as useful in the infinite dimensional topological space case as it is in the finite dimensional case, you can see less emphasis on what the exact dimension is in this case.
However, in this situation you get into discussions of other types of coordinate systems (such as Schauder bases, FDDs, unconditional bases, etc.), which are different from the notion of an (algebraic) basis. You also can ask about density character instead of dimension, which is the smallest cardinality of a dense subset. This encodes topological information, while the purely algebraic notion of a basis does not. For example, infinite dimensional Hilbert space $\ell_2$ has no countable basis, it does have a countable, dense subset. So the dimension is that of the continuum, but the density character is $\aleph_0$.