We consider linear bounded operators acting in Banach spaces with a basis, such operators can be represented by an infinite matrix. We prove that for an invertible operator there exists a sequence of invertible finite-dimensional operators so that the family of norms of their inverses is uniformly bounded. It leads to the fact that solutions of finite-dimensional equations converge to the solution of initial operator equation with infinite-dimensional matrix.

翻译：我们考虑在具有基的巴拿赫空间中作用的有界线性算子，此类算子可由无穷矩阵表示。我们证明，对于可逆算子，存在一列可逆有限维算子，使得其逆的范数族一致有界。这一结论导致有限维方程的解收敛于具有无穷维矩阵的原始算子方程的解。