• Hahn-Banach Theorem (including geometric versions)
  Applications: Extensions of positive functionals, Banach limits, and finitely additive measures
• Normed linear spaces (their equivalence and basic operations: quotients, … )
• Banach spaces (l^p, L^p, W^k,p, C(X), B(X), M(X), c₀ …, the issue of separability)
• Equivalence of finite-dimensional Banach spaces
• The unit ball: its non-compactness, strict and uniform convexity (and the problem of distance minimalization)
• Isometries of Banach spaces (Mazur-Ulam's theorem on linearity)
• Fixed point theorems of Banach, Brouwer, and Schauder
  Applications: existence for integral and differential equations
• Symmetric (sesquilinear) bilinear forms, inequalities of Schwarz and Minkowski, the parallelogram identity and polarization
• Hilbert spaces and the squared distance minimization
• Complete orthonormal sets and classification of Hilbert spaces
• Fourier transform as an isometry (Parseval and Plancherer identities)
• Riesz-Frechet representation theorem (and its Lax-Milgram extention)
  Applications: Radon-Nikodym Theorem, solution existence for PDE's
• Dual spaces, reflexive and non-reflexive spaces (with examples)
• Applications: Runge's Theorem, existence of Green's functions
• Weak convergence (and examples in concrete spaces, e.g., weak=strong in l¹)
• Baire category and the uniform boundedness principle (weak sequential equals strong closure of convex sets)
• weak* sequential compactness of the unit ball
• Applications: convolution approximation and divergent Fourier series, weak analyticity implies strong analyticity
• Locally convex topological linear spaces (weak topologies versus weak convergence) and Krein-Milman theorem
  Applications: existence of ergodic measures, Stone-Weierstrass Theorem, positive definite functions (theorems of Caratheodory and Bochner)
• Linear maps, boundedness and continuity
• Transpose (adjoint) of a linear map (and Fredholm Alternative)
• The space L(X, Y) with various topologies
• Baire category: Uniform Boundedness Principle, Open Mapping Theorem, and Closed Graph Theorem
  Applications: the well posedness principle, continuity of projections in algebraic Y ⊕ Z, finite co-dimension ranges are closed, self-adjointness implies continuity, continuous operator into range of a compact operator is compact, …
• Integral operators: general criteria for boundedness and some singular kernels: Fourier, Laplace, and Hilbert transforms
• Banach algebras (examples and the regular representation)
• Spectrum, the Neumann (geometric) series and spectral radius formula
• The point, continuous, and residual spectrum with examples (multiplication operators, lift/right shifts, the discrete laplacian, Volterra operators, …)
• Analytic functional calculus
• Gelfand's theory and the characterisation of commutative C*-algebras
  Applications: Wiener's theorem on absolute convergence of Fourier series
• Compact operators, basic properties, comparison with finite dimensional operators, examples
• The Dimension Theorem (Index=0) for compact perturbations of the identity
• Spectral theory of compact operators
• Spectral theory of compact self-adjoint operators via Relich quotients
• Spectral theorem for self-adjoint operators, cyclicity, multiplicity, and spectral measures

Texts:

Functional Analysis (the primary text), by P.D. Lax
Methods of Modern Mathematical Physics, by M. Reed and B. Simon
Applied Functional Analysis, by E. Zeidler
Functional Analysis, by F. Riesz and B. Sz.-Nagy
Introductory Functional Analysis with Applications, by E. Kreyszig