= Annihilator of a second-difference span in l2
{title2=$x_k=e_k-2re_{k+1}+e_{k+2},\quad r>1$}
Let $M$ be the closed span of these vectors in the real <l2 sequence space>. A vector $\beta$ is orthogonal to $M$ exactly when $\beta_{k+2}-2r\beta_{k+1}+\beta_k=0$. The characteristic roots are $r\pm\sqrt{r^2-1}$; one grows geometrically and the other, $\rho$, lies in $(0,1)$. Square summability eliminates the growing solution, so $M^\perp=\operatorname{span}\{(\rho^{j-1})_{j\geq1}\}$. Hence $M$ is a closed hyperplane, namely the kernel of the bounded functional $\alpha\mapsto\sum_j\alpha_j\rho^{j-1}$. This combines a recurrence with <density by annihilators>.
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