Specular differentiation in normed vector spaces: Quasi-Mean Value and Quasi-Fermat Theorems
The work provides a new theoretical framework for differentiation in normed spaces, which may be of interest to mathematicians studying functional analysis and convex analysis, but the results are purely theoretical with no immediate applications or empirical validation.
This paper introduces specular differentiation, a generalization of Gâteaux and Fréchet differentiation in normed vector spaces, and establishes weak forms of the Mean Value Theorem and Fermat's Theorem in this context. It also identifies a distinguished element of the Fréchet subdifferential of a convex function via specular differentiation.
This paper introduces specular differentiation, which generalizes Gâteaux and Fréchet differentiation in normed vector spaces. We investigate its fundamental theoretical properties and establish weak forms of the Mean Value Theorem and Fermat's Theorem in the specular sense. Finally, we identify a distinguished element of the Fréchet subdifferential of a convex function through specular differentiation.