Closed linear span ($\overline{\mathrm{span}}S$)

In a normed vector space, the closed linear span of $S$ is the closure of its linear span in the norm topology. It includes limits of finite linear combinations. In a Hilbert space, an infinite orthonormal basis spans in this closed sense, allowing norm-convergent series in addition to finite sums.

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 $M^{\perp }=\mathrm{span}\{({\rho }^{j-1}{)}_{j\ge 1}\}$. 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.