TITLE: Why is the exponential function not in the subspace of all polynomials?
QUESTION [16 upvotes]: The exponential function can be written as
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots.$$
The subspace of all polynomials is $$\text{span}\{1, x,x^2, \dots \}$$
Sure $e^x$ is in this set?

REPLY [7 votes]: $\mathrm{span}(A)$ is the set of finite linear combinations of terms from $A$. Infinite sums require notions of limits and bring up issues of convergence radii (there are plenty of infinite polynomial that converge only at a single point).