Uma introdução ao cálculo diferencial em espaços normados: regra da cadeia e desigualdade do valor médio

Abstract

This work presents a rigorous introduction to differential calculus in normed spaces, defining the Fréchet derivative and establishing normed space versions of the Chain Rule and the Mean Value Inequality. The motivation for this work lies in attempting to follow one of the essential paths for the construction of Functional Analysis, the differential calculus proposed by Fréchet. The main objective is to provide proofs of the Chain Rule Theorem and the Mean Value Inequality for normed spaces. The methodology adopted was bibliographic research and theoretical synthesis, with the collection and comparison of definitions and proofs in classical references of Real and Functional Analysis. As a result, the Chain Rule for compositions of differentiable maps in normed spaces and the Mean Value Inequality are proven, along with examples and observations on the connection with calculus in R and applications to the study of continuous linear operators. The work offers pedagogical material that facilitates the student’s transition from classical calculus to concepts of Functional Analysis.