Banach sequence spaces equipped with sequential norms (Q1825406)

Let E be a Banach space and \(\omega\) (E) the vector space of all E-valued sequences. A norm \(\rho\) : \(\omega\) (E)\(\to {\mathbb{R}}^+\cup \{+\infty \}\) is called a sequential norm if i) \(\exists m\leq M<\infty\) such that \(m.\| u\|_ E\leq \inf_{k}\rho (ue_ k)\leq \sup_{k}\rho (ue_ k)\leq M\) for all \(u\in E\) and ii) \(\rho (x)=\sup_{N}\rho (\sum^{N}_{k=1}x_ ke_ k)\), \(x=(x_ k)\in \omega (E).\) Then \(X(E)=\{x\in \omega (E):\) \(\rho (x)<+\infty \}\) is called a sequentially normed space. Given Banach spaces E and F, the \(\alpha\)- and \(\beta\)-duals of X(E) are defined by \[ X(E)^{\alpha}=\{(A_ k)\subset L(E,F);\quad \sum^{\infty}_{k=1}\| A_ kx_ k\|_ F\quad converges,\quad \forall (x_ k)\in X(E)\}, \] \[ X(E)^{\beta}=\{(A_ k)\subset L(E,F);\quad \sum^{\infty}_{k=1}A_ kx_ k\quad converges\quad in\quad F,\quad \forall (x_ k)\in X(E)\}. \] If s s is a solid sequentially normed real-valued sequence space containing \(\phi\) then s s(E)\(=\{(x_ k\}\subset E\); \(\| x_ k\| \in s s\}\) is a sequentially normed space. The author then studies the \(\alpha\)- and \(\beta\)-duals of such spaces and matrix transformations between them. The tackled problems may be compared with those treated in the case of ``ordinary'' sequence spaces.