Polynomial Operator in the Shifts in Discrete Algebraic Dynamical Systems

The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to an appropriate scalar product. It follows that the polynomial operator in the shift which U. Oberst and J. C. Willems have introduced to define time invariant discrete linear dynamical systems can be explained as the adjoint of the polynomial multiplication.