In this paper, we study a Bishop–Phelps–Bollobás type property called the property \({\mathbf{L}}_{o,o}\) of a pair of Banach spaces. Getting motivated by this, we introduce the notion of Approximate minimizing property (AMp) of a pair of Banach spaces and characterize finite dimensionality of Banach spaces with respect to this property. We further introduce the notion of approximate minimum norm attainment set of a bounded linear operator and characterize the AMp with the help of Hausdorff convergence of the sequence of approximate minimum norm attainment sets of bounded linear operators. We also investigate sufficient conditions for the holding of some weaker forms of the AMp for a pair of Banach spaces. Finally, we define and study uniform \(\varepsilon\)-approximation of a bounded linear operator in terms of its minimum norm.

在本文中，我们研究了 Bishop-Phelps-Bollobás 类型的性质，称为一对 Banach 空间的性质\({\mathbf{L}}_{o,o}\)。受此启发，我们引入了一对 Banach 空间的近似最小化属性 (AMp) 的概念，并根据该属性来表征 Banach 空间的有限维数。我们进一步介绍了有界线性算子的近似最小范数达到集的概念，并借助有界线性算子的近似最小范数达到集序列的 Hausdorff 收敛来表征 AMp。我们还研究了为一对 Banach 空间保持一些较弱形式的 AMp 的充分条件。最后，我们定义和研究统一\(\varepsilon\)- 有界线性算子在其最小范数方面的近似。