Functions in $\mathbb {R}[X] $

Question

For the ring of polynomials over the reals, which can be considered an infinite-dimensional vector space with infinite monomial basis, is the following true:

Any analytic function $f$, which is not a polynomial, is also in the vector space/ring of real polynomials?

Thanks

Answer 1

If something is not a poynomial, it can not be in the set of polynomials.

The axiom of closure under addition states that if $x$ and $y$ are two elements of a ring, then $x+y$ is also in the ring. But an axiom can not be applied infinitely many times, so this does not imply that infinite sums must be in the ring.

On the other hand, the set of analytic functions are a bigger ring, and the set of formal series are an even bigger ring.