Our definition of Garlerkin Scheme / Base is as follows:
Definition (Garlerkin Scheme / Base): Let $(V, \| \cdot \|)$ be a Banach space. Then the family $(V_n \subset V)_{n \in \mathbb{N}}$ of finite-dimensional subspaces is called Galerkin-Scheme if it is complete in the limit, i.e. \begin{align} \label{eq:aprroxerror} \tag{approximation error} \lim_{n \to \infty} \textrm{dist}(V_n,v) = 0 \quad \forall v \in V \end{align} A pairwise linearly independent sequence $(\Phi_k)_{k \in \mathbb{N}} \subset V$ is called Galerkin-Base if $\big(\text{span}((\Phi_k)_{k = 1}^{n})\big)_{n \in \mathbb{N}}$ forms a \textsc{Galerkin}-Scheme.
My Questions
- (Why) do we need $V$ to be complete?
- Why does the base only have to be pairwise linearly independent? This is a very weak condition, i.e the vectors $((1,0),(0,1),(1,1))$ are pairwise linearly independent, but don't fulfil the intuition of a base. When we show that every separable space has
1.) Completeness is the difference between a normed space and a Banach space. So if $V$ is a Banach space, it has to be complete.
You want the Galerkin scheme, the sequence of subspaces of $V$ to be complete in the defined sense to have the assurance that with sufficient refinement (large enough $n$ in $V_n$) you get approximation errors below any given tolerance.
2.) Many useful function "bases" (generating systems) are not independent in the finite or infinite sense. But you want to exclude the too trivial dependencies, like where one generator is a multiple of another. In a more strict sense one would like to demand that the optimal approximation coefficients (smallest distance, smallest size, in whatever metric) are unique.
For instance, in 2-channel wavelets symmetry and orthogonality are exclusive (in the continuous examples). But one can easily find examples that have both if one changes from one to two high-pass bands, getting a tight oversampled wavelet frame. Examples based on B-splines were constructed in Charles K. Chui, Wenjie He (2000) " Compactly supported tight frames associated with refinable functions", p.15ff