Let $Y\subset l_{\infty }^{\left(n\right)}$ be a hyperplane and let $A\in ℒ\left(Y\right)$ be given. Denote $$\begin{array}{ccc}\hfill 𝒜=& \{L\in ℒ(l_{\infty }^{\left(n\right)},Y):L\mid Y=A\}\text{and}\hfill & \hfill {\lambda }_A=inf\{\parallel L\parallel :L\in 𝒜\}.\end{array}$$ In this paper the problem of calculating of the constant ${\lambda }_A$ is studied. We present a complete characterization of those $A\in ℒ\left(Y\right)$ for which ${\lambda }_A=\parallel A\parallel$. Next we consider the case ${\lambda }_A>\parallel A\parallel$. Finally some computer examples will be presented.

In this note we investigate the relationship between the convergence of the sequence $\left\{S_n\right\}$ of sums of independent random elements of the form $S_n={\sum }_{i=1}^n{\epsilon }_i{x}_i$ (where ${\epsilon }_i$ takes the values $\pm 1$ with the same probability and $x_i$ belongs to a real Banach space $X$ for each $i\in \mathbb{N}$) and the existence of certain weakly unconditionally Cauchy subseries of ${\sum }_{n=1}^{\infty }{x}_n$.