= Geometric sequences with accumulating ratios span ell-one
{title2=$\overline{\operatorname{span}\{(1,\beta_k,\beta_k^2,\ldots)\}}=\ell^1$}
Suppose the distinct complex numbers $\beta_k$ satisfy $|\beta_k|\leq\delta<1$. The <geometric sequences> in the displayed formula belong to the <absolutely summable sequence space> and have <dense> linear span. Otherwise the <Hahn-Banach theorem> gives a nonzero annihilating functional, represented by a <bounded sequence> $b_j$. Its <power series> $F(z)=\sum_{j\geq1}b_jz^{j-1}$ is holomorphic in the unit disk and vanishes at every $\beta_k$. Compactness supplies an accumulation point inside that disk. The <identity theorem> forces $F=0$ and hence all its coefficients zero, a contradiction. This is a useful form of <density by annihilators> that converts a Banach-space problem into uniqueness of an analytic function.
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