Compositions and adjoints of unbounded operators

This lecture covers the fundamental concepts of unbounded operators, resembling the momentum operator of a particle in quantum mechanics. The algebraic manipulation of unbounded operators must be done carefully, defining the sum and product of operators. The adjoint of an operator is the next step, with no reason for it to be bounded. The lecture explores the domain of the adjoint, defining auto-adjoint and normal operators. Examples illustrate the integration by parts and the properties of symmetric and auto-adjoint operators, emphasizing the importance of closure in operator theory.