6. Appendix

Contents:

• 6.1. Differential calculus
  • 6.1.1. Basic definitions and notations
  • 6.1.2. Derivatives in normed vector spaces
  • 6.1.3. Dual spaces
  • 6.1.4. Banach spaces
  • 6.1.5. Hilbert spaces
  • 6.1.6. The implicit function theorem
• 6.2. Complements of functional analysis
  • 6.2.1. The Banach contraction mapping theorem
  • 6.2.2. The uniqueness of the limit argument
  • 6.2.3. Weak convergence in a Hilbert space
  • 6.2.4. Compactness
• 6.3. More advanced functional analysis
  • 6.3.1. Nemitski operators
• 6.4. Convexity
  • 6.4.1. Projection onto convex subsets in Hilbert spaces
• 6.5. Optimization theory
  • 6.5.1. Necessary first-order optimality conditions
  • 6.5.2. The convex case
• 6.6. PDE
  • 6.6.1. A few words about the theory of distributions
  • 6.6.2. A glance at elliptic regularity theory