My math methods textbook (Hassani's Mathematical Physics) makes the following (I think dubious) claim:
If we assume that a < 0 < b, then the set of monomials $1,x,x^2,...$ forms a basis for $C^∞(a,b)$, because, by Taylor’s theorem, any function belonging to $C^∞(a,b)$ can be expanded in an infinite power series about $x = 0$. Thus, this space is infinite-dimensional.
Now this claim seems to me dubious for two reasons:
(1) It does not use the notion of basis (I think the technical term is Hamel basis) previously defined: a basis is a linearly independent set of vectors which span the space (where span only involves finite linear combinations). $e^x$ for example is not representable as a finite linear combination of these monomials.
(2) Even if we did admit the generalized notion of basis (I think this is called a Schauder basis?) I don't think it's true that the monomials form a (Schauder) basis; as far as I know, it is not true that every element of $C^∞(a,b)$ is representable as a power series (the analytic functions are a proper subset of the smooth functions on $(a,b)$).
Am I correct in these two assessments or am I missing something? Note that I do not disagree with the statement about the space being infinite-dimensional (of course).