Discrete linear 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 a $\mathit{\text{scalar}}\mathit{\text{product}}$, which we used to define $\mathit{\text{orthogonals}}$ in these spaces. The closed subspaces in the first vector space are then the orthogonals of subsets in the second space. Using power series and polynomials, we prove that the $\mathit{\text{polynomial}}\mathit{\text{operator}}\mathit{\text{in}}\mathit{\text{the}}\mathit{\text{shift}}$ which U. Oberst and J. C. Willems have introduced to define time invariant discrete linear dynamical systems is the functorial adjoint of the polynomial multiplication. These results are generalized to the case of vectors of sequences and vectors of power series and polynomials. We end this paper by describing discrete linear algebraic dynamical systems.