A Steinhaus type theorem

Abstract:

The sequence space ${\Lambda _r},r \geq 1$ being a fixed integer, is defined as \[ {\Lambda _r} = \left \{ {x = \left \{ {{x_k}} \right \} \in {l_\infty },{x_k} \in K,k = 0,1,2, \ldots ,\left | {{x_{k + r}} - {x_k}} \right | \to 0,k \to \infty } \right \},\] where $K$ is a complete, nontrivially valued field and ${l_\infty }$ is the space of bounded sequences with entries in $K$. In this paper, it is proved that given a regular matrix $A = ({a_{nk}}),{a_{nk}} \in K = {\mathbf {R}}$ or ${\mathbf {C}}$, there exists a sequence in ${\Lambda _r} - \cup _{i = 1}^{r - 1}{\Lambda _i}$ which is not $A$-summable. This is an improvement of the well-known Steinhaus theorem. It is, however, shown that this result fails to hold when $K$ is a complete, nontrivially valued, nonarchimedean field, whereas it is known that the Steinhaus theorem continues to hold.