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Analysis in Rⁿ, Volume 04: Derivatives in Vector Spaces is an advanced mathematics textbook devoted to the rigorous study of differentiation in vector and abstract spaces.

This volume extends the classical theory of derivatives from single-variable real analysis to functions defined between normed vector spaces, abstract vector spaces, and multidimensional settings. It is designed for students, teachers, researchers, and independent learners who want to strengthen their understanding of advanced analysis.

The book develops the theory progressively through three main chapters:

Derivatives and Integrals in Vector Spaces: limits, derivatives, Riemann integration in vector spaces, and the Fundamental Theorem of Calculus.

Derivatives in Abstract Vector Spaces: Fréchet derivatives, bounded linear transformations, uniqueness of the derivative, and the chain rule in abstract settings.

Directional and Partial Derivatives: partial derivatives, directional derivatives, generalized mean value results, coordinate decomposition, and gradient-based techniques.

Each section combines definitions, theorems, proofs, examples, solved exercises, and proposed problems. The text is suitable for advanced undergraduate students, master’s degree students, and readers preparing for higher studies in mathematical analysis, functional analysis, optimization, differential geometry, and mathematical physics.

Series: Master’s Degree in Mathematics Series
Language: English
Author: Helbert Justo Luque Zevallos

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