Bounded Operators: Theory and Applications

This lecture introduces the concept of bounded operators between normed vector spaces, defining them as linear applications and exploring examples such as the position and momentum operators in quantum mechanics. The lecture discusses the importance of continuity in operators, showing that a bounded operator is necessarily continuous. It also covers the dual space of a normed vector space, explaining how bounded operators can be uniquely defined by their restriction to a dense subset of a Banach space. The extension of operators, exemplified by the Fourier transform, is presented, highlighting its properties and applications. The lecture concludes with the definition of the adjoint operator and its role in linear bounded transformations.