Program of the course Advanced Analysis –A (2023-2024)

0. Metric and normed spaces, converging and Cauchy sequences, completeness, Banach spaces. Completion theorem. Dimension, finite-dimensional spaces and infinite-dimensional spaces. Space of real polynomials on an interval; space of real continuous functions on an interval. Compactness, sequential compactness, precompactness, relative compactness and their relations. Density and separability. Linear bounded operators on Banach spaces and their basic properties. Extension theorem. Space of continuous linear operators between Banach spaces and its algebraic and metric properties. Dual space.

1. Hahn-Banach theorem in analitic form for real and complex vector spaces. First consequences of Hahn-Banach space. Convexity. Gauge of a convex set and its properties, geometric forms of Hahn-Banach theorem. Distance function and its properties. Baire theorem, Banach-Steinhaus theorem, Open mapping theorem, Inverse function theorem, Closed graph theorem. Diagonal procedure. Strong precompactness criterion in normed spaces. Convex set and convex functions. Continuity of convex functions, convex hull. Krein-Milman theorem. Examples: space of continuous functions on an interval, ℓp spaces: algebraic and metric properties.

2. Hilbert spaces; othogonality and orthonormality. Pitagorean theorem, Bessel inequality, parallelogram identity. Projection on a closed convex set, orthogonal complement, projection operators, direct sum. Riesz theorem. Reflexivity of Hilbert spaces. Orthonormalization procedure. Hilbert bases; separability and existence of a countable basis. Parseval identity. Lax-Milgram theorem.

3. Space of continuous functions on a compact metric space; completeness and separability. Partition of unity. Equicontinuity and Ascoli-Arzel\'a theorem. Dini's lemma. Density of separating lattices, separating subalgebras, Stone-Weierstrass theorem. Exercises and examples on norm, support and convergence in the dual of C(\Rd).

4. Lp spaces. Definition and basic properties. H\"older and Minkowski inequalities, separability, interpolation. Support and essential support of a function. Functions with compact support, mollifiers. Convolution, Young's inequality, density of smooth functions in Lp. Strong compactness Kolmogorov criterion. Duality and Riesz theorem. Weak convergence in Lp, weak* convergence in L∞ and their characterization. Riemann-Lebesgue lemma. Clarkson inequalities and uniform convexity of Lp spaces. Examples and exercises.

5. Topology generated by a family of functions. Weak topology in a Banach space. Weakly converging sequences. Weak and strong closure, Mazur lemma. Bidual space and reflexive spaces. Strong and weak continuity of linear operators. Weak* topology. Banach-Alaoglu theorem. Helly's lemma, weak* density of the unit ball in the bidual. Kakutani theorem and its consequences. Reflexivity of a space, of its dual and of its closed subspaces. Sequential relative compactness in reflexive Banach spaces. Minimum theorem for sequential weakly lower semicontinuous functionals. Uniform convexity; Milman theorem, weak-strong convergence in uniformly convex spaces.

6. Main properties of the algebra of continuous linear operators on Banach spaces. Composition, invertibility, power of an operator. Inverse operator and its properties, formula of the inverse. Resolvent set and spectrum of an operator. Resolvent operator and resolvent formula. Properties of the spectrum, spectral radius. Operators on Hilbert spaces. Adjoint operator, selfadjoint operators and their properties. Spectral properties of the adjoint operator and of compact operators. Compact selfadjoint operators and their spectral properties. Spectral decomposition.

Textbooks:

[1] V. Checcucci, A. Vesentini, E. Tognoli, Lezioni di topologia generale, Feltrinelli.

[2] R.L. Wheeden, A. Zygmund, Measure and Integral, Taylor-Francis.

[3] M. Reed, B. Simon, Functional Analysis, Academic Press.

[4] H. Brezis, Functional Analysis, Sobolev spaces and PDE's. Springer.

[5] F. Hirsch, G. Lacombe, Elememts of Functional Analysis. Springer