ABSTRACT
From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space. This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products, and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space. This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics.
