Functional Analysis and Differential Equations in Abstract Spaces

Part 1 Differential equations in abstract space: Abstract Cauchy problem and associated semigraphs; uniqueness of the abstract Cauchy problem in Hilbert and Banach spaces; bounded solutions on the real line; almost-periodic solutions; ultraweak solutions (local existence and approximation); regularity of solutions. Part 2 Functional analysis: Hilbert space, definitions, first results; geometry in Hilbert spaces; duality, orthonormal bases; normed and Banach spaces, examples; linear continuous operators; duality, Hahn-Banach theorem; compact linear operators; uniform boundedness theorem - open mapping theorem, Hellinger-Toeplitz theorem, adjoint operators, closed graph theorem; bounded symmetric operators in Hilbert spaces - square root, orthogonal proejctions, spectral theorem, spectrum; unbounded operators in Banach and Hilbert spaces - definition, exmaples, adjoints; closed and closable operators; symmetric unbounded operators in Hilbert space - adjoint, self-adjoint operators; the classical abstract Cauchy problem - semigroups of class Co; spectral theory for unbounded operators; sesquilinear forms and associated operators.