Spectral Representations of Vectors and Operators

Let be a vectorial (linear) finite space with scalar (dot) product operation included, V /C, with the (ortho-normal) basis or vectors  

Within this representation the scalar product may be written as  

Moreover, the last operator constitutes the link between the vector and operatorial representation, i.e., representing an operator (the unity one) written in terms of bra- and ket-vectors or the Hilbert space. 

Before proceeding further worth noting that from the inner structure of the Hilbert space it can be completed in the sense of norm 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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