# Definition:Complete Ritz Sequence

## Definition

Let $\mathcal M$ be a normed linear space.

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

Let $\mathcal M_n$ be an $n$-dimensional linear subspace of $\mathcal M$, 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:

$\forall y\in\mathcal M:\forall\epsilon>0:\exists \map n {\epsilon}\in\N:\exists\eta_n\in\mathcal M_n:\size {\eta_n-y}<\epsilon$

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

## Source of Name

This entry was named for Walther Ritz.