As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are **no** non-trivial polynomials in that space and, to me, polynomials are not only the simplest functions, they are the building blocks of most everything which can be (easily) manipulated algorithmically. And *restricting to a compact support* is really a non-answer, since one of the great things about polynomials is that they are global, analytic functions.


To ask a more precise question: are there some spaces of (total, real-valued) functions which are both nice from a functional analysis point of view, and contain all the polynomials?



Polynomials and L^p(R)
You are referring to $L^p(\mathbb{R}, \mathcal{B}, \mu)$ in the case that $\mathbb{R}$ is endowed with Lebesgue measure $\mu$. Consider instead the measure $\nu$ given by $d\nu = f d\mu$, where $f$ is in the [Schwartz space](http://en.wikipedia.org/wiki/Schwartz_space) and $f$ does not take the value zero. Because the product of a polynomial with $f$ is also in the Schwartz space, and the Schwartz space is contained in $L^p(\mathbb{R}, \mathcal{B}, \mu)$, it follows that polynomials are in $L^p(\mathbb{R}, \mathcal{B}, \nu)$.


