New-Maths
Mathematics advances by changing the admissible carrier of invariance. New-Maths Architecture TOC Book trajectory: exact objects → completed objects → topological control → measurable size → measurable functions → integration → differentiation almost everywhere → Banach/function-space structure → duality → distributions → Sobolev weak derivatives → weak PDE regularity. Preface The preface gives the book’s architecture: real analysis is not just limits and integrals. It is a sequence of representational upgrades. Measure theory is developed first; then functional analysis and linear functionals; then distributions; then Sobolev functions; finally weak harmonic functions. The major theme is that rough objects become tractable when re-expressed through measures, linear functionals, test functions, and weak derivatives. Chapter 1. Preliminaries This chapter installs the basic formal substrate: sets, relations, functions, equivalence relations, and choice princi...