Conditional Bases in the Matrix Space with the Cut-Norm

In the paper we consider the space $\mathcal{L}_{0,1}$ of infinite matrices $(a_{ij})_{i,j=1}^\infty$, where $a_{ij}\in\mathbb{R}$, equipped with a cut-norm. The cut-norm of a matrix appears naturally in various areas of discrete mathematics, and is also used in functional analysis. It is equivalent to the operator norm of a matrix considered as a mapping from the space $c_0$ of sequences converging to zero (or from $l^\infty$) to $l^1$. It is proved that the set of matrix units $\{E_{ij}\}_{i,j=1}^\infty$ forms a system of random unconditional convergence (RUC) in the space $\mathcal{L}_{0,1}$, but is not a system of unconditional convergence. This system is linearly full in $\mathcal{L}_{0,1}$ and, being enumerated in a certain order, may turn out to be a conditional basis. We obtain both necessary and sufficient conditions for a bijection $\ell\colon \mathbb{N}\to \mathbb{N}^2$, under which the sequence $(E_{\ell(k)})$ forms a basis in $\mathcal{L}_{0,1}$. In~particular, any successive filling of the corner squares leads to a basis, while the triangular filling not. It~is~shown that random permutations of the sequence $(E_{\ell(k)})$ almost surely do not form a basis. The problem of a complete characterization of all bijections $\ell\colon \mathbb{N}\to \mathbb{N}^2$ that lead to the basis $(E_{\ell(k)})$ is set up, and it is constructively shown that the proposed necessary and sufficient conditions do not provide a~complete answer to this problem.