TITLE: Does $\text{Span}(\{1, e^x, e^{2x}, . . . e^{nx}, . . .\})=C(\mathbb{R})$?
QUESTION [9 upvotes]: Does the set of functions $A = \{1, e^x, e^{2x}, . . . e^{nx}, . . .\}\subseteq C(\mathbb{R})$ span $C(\mathbb{R})$? Here $C(\mathbb{R})$ is
the vector space of all continuous functions $f : \mathbb{R} \rightarrow \mathbb{R}$. Will the answer change if we
consider $A$ as a subset of the vector space of all differentiable functions from $\mathbb{R}$ to $\mathbb{R}$?
I think the answer to the first question is no, and I think that $f(x)=2^x$ is a counterexample. I am not sure about the second question. Any help is appreciated, thank you.

REPLY [4 votes]: The span of the functions $e^{nx}$, $n\in{\mathbb N}_{\geq0}$, consists of the finite linear combinations of these functions, and not a single function more than that. In other words, it consists of the functions $x\mapsto p\bigl(e^x\bigr)$, with $p$ an arbitrary polynomial of one variable.