Definition:Complete Ritz Sequence

Definition
Let $\MM$ be a normed linear space.

Let $\sequence {\phi_n}$ be a Ritz sequence in $\MM$.

Let $\MM_n$ be an $n$-dimensional linear subspace of $\MM$, spanned by the first $n$ mappings of $\sequence {\phi_n}$.

Let $\eta_n$ be of the form:


 * $\eta_n = \boldsymbol \alpha \boldsymbol \phi$

where $\boldsymbol \alpha$ is an $n$-dimensional real vector.

Suppose:


 * $\forall y \in \MM: \forall \epsilon > 0: \exists \map n \epsilon \in \N: \exists \eta_n \in \MM_n: \size {\eta_n - y} < \epsilon$

Then the sequence $\sequence{\phi_n}$ is called complete in $\MM$.