Section 1.7 Outline
Chapters Chapter 3, Chapter 4, and Chapter 5 contain classical introductory material from functional analysis. We introduce Banach spaces and Hilbert spaces, talk about different norms, and define operators between these spaces. We learn when an operator is bounded, continuous, closed, compact, or symmetric, and we prove the Uniform Boundedness Principle. The introduction of dual spaces, and the important Hahn-Banach theorem, bring additional structure to the function spaces, which allow us to distinguish between weak and weak\(^*\) convergence. An important consequence is the Alaoglu weak\(^*\) compactness result, which we prove in Chapter Chapter 5. Chapters Chapter 3-Chapter 5 contain essential background that should be studied by readers that are new to this area. These chapters can be skipped by experienced functional analysts. Chapter Chapter 6 on Sobolev spaces makes a distinct jump towards PDEs. General solutions of PDEs often only allow weak derivatives, which we define from a distributional derivative. Sobolev spaces collect functions with weak derivatives into a ranking of increased regularity. The Sobolev embeddings make explicit relations between Sobolev spaces, spaces of integrable functions and spaces of continuous and differentiable functions. We present these relationships in the Rainbow of function spaces where we show the relations between smooth functions, H\"older continuous functions, continuous and differentiable functions, integrable functions, measures, Sobolev spaces and their duals. The Rainbow of function spaces is a real highlight of this text, as it gives the reader a visual tool to better understand the relationships between all these spaces. Chapter Chapter 7 on Fixed Point Theorems steers us into very traditional mathematics. Based on topological arguments (e.g. Brouwers fixed point theorem), we develop standard results such as the Banach fixed point theorem and the Schauder fixed point theorem. The Leray-Schauder principle again makes a connection to PDE theory. It shows that a-priori estimates can lead to fixed points. Finally, the Lax-Milgram theorem, is not really a fixed point theorem, but it fits into this chapter as it ensures the existence of solutions to bilinear operator equations, as they appear in solution theory of PDEs. Variational Calculus in Chapter[cross-reference to target(s) "c-variations" missing or not unique], has developed from optimization of mechanical systems, specifically for minimizing the underlying energies. Consequently, we begin Chapter [cross-reference to target(s) "c-variations" missing or not unique] with two classical mechanical problems, the hanging chain and the rolling ball. We find that the language of variational calculus is functional analysis, hence the tools developed so far allow us to formulate optimization problems in a systematic way. We derive the first variation, the Euler-Lagrange equations and the second variation. We discuss Hamiltons principle and we derive conditions such that a minimizer exists. In Chapter Chapter 8 on Spectral Theory we come back to the analysis of operators. Where matrices have eigenvalues, linear operators in general have spectral values which include eigenvalues (point spectrum) plus elements of the continuous and residual spectra. We introduce a new best friend, which is the resolvent \(R_\lambda(A)\text{.}\) The resolvent is needed to identify the different parts of the spectrum. Also, we relate the spectrum of an operator to the spectrum of the adjoint and we formulate some spectral theorems, including the Fredholm alternative. With Spectral Theory under our belt, we develop in Chapter Chapter 9 the Semigroup Theory. Semigroup theory is the framework in which an operator exponential \(T(t)= e^{tA}\) will be defined. It can be understood as a solution of an abstract differential equation \(u_t = Au\) in a Banach space. The two main theorems in this context are the Hille-Yosida theorem and the Lumer-Phillips theorem, which we prove in Chapter Chapter 9. Based on the spectrum of so called "sectorial" operators, we will define analytic semigroups. These are important in many applications since the Laplacian \(\Delta\) generates analytic semigroups in suitable domains. The semigroup theory is build on intricate connections between the generator \(A\text{,}\) the semigroup \(T(t)\) and the generator \(R_\lambda(A)\text{.}\) To illustrate these relations, we use the Semigroup Triangle as a visual tool.