Let $C([0,1], \mathbb R)$ denote the space of real continuous functions at $[0,1]$, with the uniform norm. Is the set $H=\{ h:[0,1] \rightarrow \mathbb R : h(x)= \sum_{j=1}^n a_j e^{b_jx} , a_j,b_j \in \mathbb R, n \in \mathbb N \}$ dense in $C([0,1], \mathbb R)$?
Answer
Yes, by the general Stone-Weierstrass Approximation Theorem. Indeed, $H$ is an algebra of continuous real-valued functions on the compact Hausdorff space $[0,1]$, and $H$ separates points and contains a non-zero constant function. That is all we need to conclude that $H$ is dense in $C([0,1],\mathbb{R})$.
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