Hilbertian Bounded Operators

This lecture covers the importance of linear applications between Hilbert spaces in quantum physics, introducing the concept of Hilbert operators. It explains the Riesz-Fréchet theorem, the identification of the dual of a Hilbert space, and the isometric anti-linear isomorphism. The lecture also delves into the properties of Hilbert operators, such as being auto-adjoint, projectors, unitary, and normal operators. It discusses C*-algebras, their properties, and exemplifies with the Fourier transform. The extension of the adjoint and the unitarity of the Fourier transform are also explored.