This lecture covers the postulates of Quantum Mechanics, focusing on the state of a system as a complex-valued vector in a Hilbert space. It delves into linear operators, eigenvalues, eigenvectors, and the concept of completeness. The lecture also discusses Dirac notation, inner products, adjoint operators, and the 'bra-ket' notation. It emphasizes the importance of self-adjoint operators and provides mathematical notations for describing physical states.