When do exponential functions form a basis?

Question

I understand that exponential functions of the form $e^{a_{i}x}; i=1,2,..,n$ and $a_{i}s$ are real and distinct are linearly independent. Do these form the basis of vector space of continuous functions from $R$ to $R$.

I am only a beginner in liner algebra. Kindly excuse if this is a silly question. My intuition about the above question is that $e^{x}$ is not defined at $\infty$ and hence can't be the basis of $R$. Am I right.?

Answer 1

$e^{x^{2}}$ cannot be expressed in the form $ \sum\limits_{k=1}^{n} c_ie^{a_ix}$.

Proof: let $M$ be the maximum of $|a_i|$. Multiply both sides by $e^{-Mx}$ and let $x \to \infty$. LHS tends to $\infty$ whereas RHS is bounded.

Answer 2

The space of continuous functions has infinite dimension (consider, for example, the set $\{1,x,x^2,\ldots,x^n,\ldots\}$ which is l.i.), so a finite subset can never be a basis.