Functional Analysis

• Preliminary elements: vector spaces and metric spaces.
• Banach spaces: basic concepts and examples. Classical sequence spaces.
• Properties of Banach spaces. Finite dimensional spaces: norm equivalence, Riesz’s lemma, compactness and finite dimension.
• Hilbert spaces: basic concepts and examples, orthogonality, orthonormal families, bases.
• Linear operators:bounded linear operators in Banach spaces, the duality properties of a Banach space, the duality properties of a Hilbert space, bounded linear operators in Hilbert spaces.
• Fundamental theorems (principles) of the Banach space theory: Hahn – Banach theorem, analytical and geometric form, separation theorems. Uniform bounded principle, open function  theorem, closed graph theorem, Banach-Steinhaus Theorem.
• Autopathy and separability. Banach spaces and Banach quotient spaces.

Weak and weak* convergence: weak convergence and weak* convergence of sequences in Banach and Hilbert spaces, bounded  and weakly bounded sets in Banach and Hilbert spaces

• Functional Analysis , F.Riesz and Nagy , Dover publications 1990.
• Functional Analysis : an introduction , Y.Eidelman,V.Milman , A.Tsolomitis , Graduate Studies in Mathematics , Volume 66 ,  2004.