Definition:Complete Ritz Sequence

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

Let $ \{ { \phi_n } \} $ be an infinite sequence of mappings in $ \mathcal M $.

Let $ \mathcal M_n $ be an $ n $-dimensional linear subspace of $ \mathcal M $, spanned by the first $ n $ mappings of $ \{ { \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:


 * $ \displaystyle \forall y \in \mathcal M : \forall \epsilon > 0 : \exists n \left ( { \epsilon } \right ) \in \N : \exists \eta_n \in \mathcal M_n : \left \vert \eta_n - y \right \vert < \epsilon $

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