Representing systems of subspaces (Q1087778)

Let \((H_ k)^{\infty}_{k=1}\) be sequence of linear subspaces of a topological linear space \(H\) over the field \(\Phi\). One says that \((H_ k)\) is a representing system for \(H\) if every \(x\in X\) can be represented as a convergent series \[ x=\sum^{\infty}_{k=1}y_ k,\quad y_ k\in H_ k,\quad k=1,2,\ldots,\tag{*} \] When \(H\) is a separated locally convex space and the above series is absolutely convergent one says that \((H_ k)\) is an absolutely representing system. The aim of the paper is to give examples of representing systems which neither satisfy \(H_ n=\Phi x_ n\), \(\forall n\), nor the representation (*) is unique (i.e. \((H_ k)\) is a basis for \(H\)). All the examples are given in the space \(H(\mathcal G)\) of analytic functions on the domain \(\mathcal G\subset \mathbb C=\Phi\) endowed with the topology of uniform convergence on compact subsets of \(\mathcal G\).