Functional Analysis

Syllabus

• Hilbert spaces: inner products, semi-inner products, the Cauchy–Bunyakovsky–Schwarz inequality, and examples such as ℓ² and L² spaces.
• Orthogonal decompositions and distance from subspaces; Riesz's theorem on decomposition into orthogonal components.
• Orthonormal systems in Hilbert spaces; generalized Fourier expansions, Bessel's inequality, Parseval's identity, and isometric isomorphisms of Hilbert spaces.
• Continuous linear functionals on Hilbert spaces; the Riesz representation theorem.
• Continuous linear operators on Hilbert spaces: operator norm, numerical radius, adjoint operator, and spectrum.
• Special classes of operators: self-adjoint, unitary and normal operators, and orthogonal projections.
• Compact operators on Hilbert spaces; the Hilbert–Schmidt theorem for normal compact and self-adjoint compact operators as an infinite-dimensional analogue of the principal axis theorem for matrices.
• Banach spaces: distance from subspaces and Riesz's lemma.
• Baire category theorem and its role in functional analysis.
• Continuous linear functionals on normed spaces; the Hahn–Banach theorem, applications, and the Mazur–Orlicz theorem.
• Continuous linear operators on Banach spaces: adjoint operators, spectrum, and Neumann series.
• Banach–Steinhaus theorems: uniform boundedness and pointwise convergence principles.
• Open mapping theorem and closed graph theorem for Banach spaces.
• Compact operators on Banach spaces; basic elements of Riesz–Fredholm theory, Schauder's theorem, and Lomonosov's theorem.