Uniform convergence and the Hahn-Schur theorem (Q2915803)

Let \(\lambda \) be a scalar sequence space containing \(c_{00}\), the space of all finite sequences. It is assumed that \(\lambda \) is either \(c_{0}\)-factorable (i.e., each \(t\in \lambda \) has a representation \(t=su:=( s_{j}u_{j})\) with \(s\in c_{0}\) and \(u\in \lambda \)) and monotone (i.e., \(vt\in \lambda \) for each \(v\in \left\{ 0,1\right\} ^{\mathbb{N}}\) and \(t\in \lambda \)) or has \(\infty \)-GHP. By definition, a sequence space \( \lambda \) has \(\infty \)-GHP (the infinite gliding hump property) if for every \(t\in \lambda \) and any increasing sequence \((I_{j}) \) of intervals, there exist an index sequence \(\left( n_{j}\right) \) and \( a_{n_{j}}>0\) with \(a_{n_{j}}\rightarrow \infty \) such that every subsequence of \((n_{j})\) has a further subsequence \((p_{j})\) such that the coordinatewise sum \(\sum_{j=1}^{\infty }a_{p_{j}}\chi _{I_{p_{j}}}t\) is in \(\lambda \) (here \(\chi _{I}\) is the characteristic function of \(I\subset \mathbb{N}\)). A series \(\sum_{j}z_{j}\) in a topological vector space is said to be \(\lambda \)-multiplier convergent if the series \(\sum_{j}t_{j}z_{j}\) converges for all \(t=\left( t_{j}\right) \in \lambda\).NEWLINENEWLINEThe author establishes some abstract results on uniform convergence of multiplier convergent series in a very general setting. Let \(E\) be a vector space, \(F\) a set and \(G\) a Hausdorff locally convex space. Let \(b:E\times F\rightarrow G\), \(\left( x,y\right) \mapsto b\left( x,y\right) =:x\cdot y\) be a map such that \(x\mapsto x\cdot y\) is linear and let \(w\left( E,F\right) \) be the weakest topology on \(E\) for which \(x\mapsto x\cdot y\) is continuous for every \(y\in F\).NEWLINENEWLINETheorem 9. Suppose that \(\sum_{j}x_{ij}\) is \(\lambda \)-multiplier convergent with respect to \(w\left( E,F\right) \) for all \(i\in \mathbb{N},\) let \( B\subset F\) be pointwise bounded (i.e., \(\left\{ b\left( x,y\right) \mid y\in B\right\} \) is bounded in \(G\) for every \(x\in E\)). If for every \(t\in \lambda \) the subset \(\left\{ \sum_{j}t_{j}x_{ij}\cdot y\mid i\in \mathbb{N} \text{, }y\in B\right\} \) is bounded in \(G\), then for every \(t\in \lambda \) the series \(\sum_{j}t_{j}x_{ij}\cdot y\) converges uniformly for \(i\in \mathbb{ N}\) and \(y\in B\).NEWLINENEWLINETheorem 14 (Hahn-Schur). Suppose that \(\sum_{j}x_{ij}\) is \(\lambda\)-multiplier convergent with respect to \(w\left( E,F\right) \) for \(i\in \mathbb{N}\), assume that \(\lim_{i}\sum_{j}t_{j}x_{ij}\cdot y\) exists for all \(t\in \lambda \) and \(y\in F\). Let \(B\subset F\) be pointwise bounded. If \( \left( a\right) \) for every \(t\in \lambda \) the subset \(\left\{ \sum_{j}t_{j}x_{ij}\cdot y\mid i\in \mathbb{N}\text{, }y\in B\right\} \) is bounded in \(G\) and \(\left( b\right) \) for each \(j\in \mathbb{N}\) there exists \(x_{j}\in E\) such that \(\lim_{i}x_{ij}\cdot y=x_{j}\cdot y\) uniformly for \(y\in B\), then \(\sum_{j}x\) is \(\lambda \)-multiplier convergent with respect to \(w\left( E,F\right) \) and \(\lim_{i}\sum_{j}t_{j}x_{ij}\cdot y=\sum_{j}t_{j}x_{j}\cdot y\) uniformly for \(y\in B\).NEWLINENEWLINEThese general results are applied in the following situations:NEWLINENEWLINE\(\left( i\right) \) \(b:L\left( X,Y\right) \times X\rightarrow Y\) is the bilinear map defined by \(b\left( T,x\right) :=Tx\) (\(X\) and \(Y\) are Hausdorff locally convex spaces), \newline \(\left( ii\right) \) \(E\) and \(F\) are vector spaces in duality and \(b:E\times F\rightarrow \mathbb{K}\) is the bilinear form, \newline \(\left( iii\right) \) \(b:C_{X}\left( S\right) \times S\rightarrow X\), \( b\left( f,s\right) :=f\left( s\right) \), where \(S\) is a compact Hausdorff space and \(X\) is a normed space, and \newline \(\left( iv\right) \) \(E\) is a topological vector space with a Schauder basis \( \left\{ b_{j}\mid j\in \mathbb{N}\right\} \), \(F:=\left\{ P_{k}\mid k\in \mathbb{N}\right\} \) and \(b:E\times \left\{ P_{k}\mid k\in \mathbb{N} \right\} \rightarrow E\) with \(b\left( x,P_{k}\right) :=P_{k}\left( x\right) , \) where \(P_{k}\left( x\right) :=\sum_{j=1}^{k}f_{j}\left( x\right) b_{j}\) (\( f_{j}\) are the coordinate functionals).