What would be the "Size" of this space be for which we have the following bases We are talking about bases in class and I am wondering if you have the bases for
$ P_\infty $ such that the bases are $$ P_0 = 1, P_1=x, P_2=x^2, ... $$
essentially the x-terms in the Taylor series representation. How many functions would be in this space $P_\infty$ ? Like would it cover all continuous functions, analytical functions, etc. ? Essentially I am wondering what would be, I guess you could say, the upper limit of this space ? I am just asking this for purely intuition and expansion of knowledge essentially... this came to me as I was reading the chapter for bases. 

The resultant space is similar to that of analytic functions, which are precisely those that can be written in the form
$$
f(x) = \sum_{i=0}^{\infty} a_n (x-x_0)^n
$$
However, around the point of expansion $x_0$, there is only a certain radius of convergence. Thus your space is one that contains many functions that do not converge anywhere but the origin, and those that do may only be defined on some interval around the origin, but where they converge they are analytic. 
