Definition:Complete Ritz Sequence

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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:


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


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

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

Source of Name

This entry was named for Walther Ritz.