# Definition:Complete Ritz Sequence

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## 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.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 8.40 $: The Ritz Method and the Method of Finite Differences