The phrase "...with applications" is not an afterthought; it is the raison d'être. Here are three domains where linear and nonlinear functional analysis converge.
Download a legitimate copy of Ciarlet’s or Brezis’s book from your university’s portal. Open to Chapter 1 (normed spaces) and Chapter 6 (nonlinear operators). Work through them in parallel. Within weeks, the infinite-dimensional world will feel as natural as ( \mathbbR^n ).
Ciarlet connects abstract theory to concrete problems, particularly in: Linear and Nonlinear Functional Analysis with Applications
Other classic texts, such as Zeidler's Nonlinear Functional Analysis and Its Applications (five volumes), similarly begin with linear tools before ascending to monotone operators and variational inequalities.
provides the flesh and blood. It captures the complexities of the real world—the friction, the curvature, and the singularities. It teaches us that even when we cannot write down a formula for the answer, we can prove the answer exists, and sometimes, that is enough to change the world.
: It covers core topics from linear functional analysis (normed vector spaces, Banach and Hilbert spaces) and transitions into nonlinear analysis (calculus of variations, Brouwer and Leray-Schauder degree theory).
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