Make an uncountable dimension vector space out of polynomials? We know that the set of all polynomials $a + bx + cx^2 +...$ forms a countably infinite dimension vector space. 
However, what if we want to form a vector space out of $f(x)=\sum_\alpha c_\alpha x^\alpha$, with $\alpha \in \mathbb R$. 
It would be the set of polynomial functions with polynomials that all have exponents within an interval.
We could then define a real function $c(x)$ that gives the values of the coefficients of the polynomials.
For example, if $c(\pi)=6$, then $f(x) = 6\cdot x^\pi + \sum_{\alpha \neq \pi}c_\alpha x^\alpha$, and so forth.


*

*Does this work properly?  

*Has this been done before? 

*Are there interesting applications or results from this?
 A: There is the field of (formal) Hahn series. The trick to a fruitful definition is you need to constrain how many nonzero coefficients you have, so that you can define multiplication by
$$ \left( \sum_\alpha c_\alpha x^\alpha \right)
 \left( \sum_\beta d_\beta x^\beta\right) 
=
\sum_{\mu} \left( \sum_\nu c_\nu d_{\mu - \nu}\right) x^\mu
$$
The problem to having a good definition is for all of the sums
$$ \sum_\nu c_\nu d_{\mu - \nu} $$
to be well-defined. The definition of Hahn series arranges it so that you're guaranteed this sum only has finitely many nonzero terms, so that you can define the uncountable sum to simply be the sum of those finitely many terms.

Of course, you can achieve your aims without being so bold — you can instead consider the ring you get by requiring a series to have only finitely many nonzero coefficients.
The result of this is pretty clearly a vector space with a basis indexed by the set of possible exponents. If you take the exponents in the reals as you suggest, it will be uncountably infinitely dimensional.
A: This is perfectly alright: Let me make an alternative uncountable dimensional vector space which is more natural (in the sense that the elements are actually  familiar real-valued functions that can be evaluated).
Let $V$ be the set of real-valued functions that are defined on the whole real line except (an unspecified) finite subset. (This set where it is undefined is allowed to vary with each function).
Clearly this is a vector space.
For a given real number $\alpha$ consider the function $f(x)= \frac1{x-\alpha}$, it is defined on the whole real line except at $\alpha$.
For any set of distinct real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$, the corresponding functions $1/(x-\alpha_1), 1/(x-\alpha_2),\ldots,1/(x-\alpha_n)$ are easily checked to be linearly independent elements of $V$. Thus $V$ is an uncountable dimensional vector space. (This has been considered by Kaplansky in his proof of of Hilbert's Nullstellensatz. See Artin's ALGEBRA).
