Paper052_Linear_Algebra_Done_Analysis

We prove that every concept in mathematical analysis—limits, continuity, differentiation, integration, measure theory, functional analysis, PDEs, distribution theory, spectral theory of unbounded operators—is a finite linear algebraic computation equipped with a convergence guarantee. The computation is the content; the guarantee is the scaffolding. The Cathedral's retreat from algebra into analysis—"your reductions are merely algebraic; the real rigor lives in analysis"—is shown to be circular: analysis was invented to provide existence proofs for limits of algebraic computations that were already being performed. Every ε-δ proof is a statement that a sequence of finite matrix approximants converges. Every Lebesgue integral is a limit of finite sums. Every Sobolev space is a completion of a space of finite linear combinations. Every spectral measure is a limit of finite spectral decompositions. The "infinite-dimensional" theory adds no new operations—it adds only the promise that the finite operations do not diverge. Analysis is linear algebra's insurance policy. The policy is not the house.