Polynomial asymptotics of solutions of differential equations

Necessary and sufficient conditions are given which are satisfied by the asymptotically polynomial solutions of normal systems of ordinary differential equations. Conditions on the right-hand side of an asymptotically linear normal systems are given under which all solutions asymptotically or strongly asymptotically approach polynomials of at most a given degree.
The concept of an almost normed space is introduced. It turns out that in a space of functions that asymptotically (and also strongly asymptotically) approach polynomials of at most a given degree one can introduce an almost norm such that in the metric it gives rise to the space is complete. An embedding theorem is given for spaces of functions that asymptotically and strongly asymptotically approach polynomials.