# Definition:Complete Ritz Sequence

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## 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 $\MM$.

## 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