We study the concepts of orthogonality and smoothness in normed linear spaces, induced by the derivatives of the norm function. We obtain analytic characterizations of the said orthogonality relations in terms of support functionals in the dual space. We also characterize the related notions of local smoothness and establish its connection with the corresponding orthogonality set, which is analogous to the well-known relation between the Birkhoff–James orthogonality and the classical notion of smoothness. The similarities and the differences between the various notions of smoothness are illustrated by considering some particular examples, including \( {\mathbb {K}}({\mathbb {H}}), \) the Banach space of all compact operators on a Hilbert space \( {\mathbb {H}}. \)

我们研究了由范数函数的导数引起的范数线性空间中的正交性和平滑度的概念。我们根据对偶空间中的支持函数，获得了上述正交关系的解析特征。我们还表征了局部光滑度的相关概念，并建立了它与相应正交性集的联系，这类似于Birkhoff-James正交性和经典光滑度概念之间的众所周知的关系。的相似性和平滑性的各种概念之间的差异，通过考虑一些特定实例，包括示出的\（{\ mathbb {K}}（{\ mathbb {H}}），\）上的所有小型运营商的Banach空间上希尔伯特空间\（{\ mathbb {H}}。\）