Function vs. Polynomial Space I've been reading up on spaces and was wondering if there was a difference between those two terms? Intuitively it would seem they are the same, but just so I don't dig myself into a hole, I was wondering if anyone had some definitive differences or it's just two terms to represent the same thing.
Thanks
 A: Functions are much (much much) more general than polynomials. A polynomial function is any function, let's say $\mathbb R \to \mathbb R$ just to fix something, which is of the form $p(x)=\sum_{k=0}^m\alpha _k x^k$, where $\alpha_0,\ldots, \alpha_m \in \mathbb R$ are called the coefficients of the polynomial function. 
Every polynomial function is continuous but there are many continuous functions which are not polynomials, for instance $f(x)=e^x$. Further still, there are many functions which are not continuous at all. The set of all polynomials is a vector space of countable dimension. The set of all continuous functions is a vector space of uncountable dimension. The set of all functions is an even larger vector space. 
One very important fact relating polynomial functions and arbitrary continuous functions on a closed interval is Weierstrass' Theorem: The polynomial functions on a closed interval $[a,b]$ are dense in the space of all continuous functions on the same interval. Thus, polynomials can be used to approximate any continuous function on a closed interval. 
