A matrix lower bound is defined that generalizes ideas apparently due to S. Banach and J. von Neumann. The matrix lower bound has a natural interpretation in functional analysis, and it satisfies many of the properties that von Neumann stated for it in a restricted case. Applications for the matrix lower bound are demonstrated in several areas. In linear algebra, the matrix lower bound of a full rank matrix equals the distance to the set of rank-deficient matrices. In numerical analysis, the ratio of the matrix norm to the matrix lower bound is a condition number for all consistent systems of linear equations. In optimization theory, the matrix lower bound suggests an identity for a class of min-max problems. In real analysis, a recursive construction that depends on the matrix lower bound shows that the level sets of continuously differential functions lie asymptotically near those of their tangents.
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