Imbedding theorems for Banach spaces of infinitely differentiable functions

Algebraic conditions are obtained for imbedding of the spaces
$$ W^{\infty}\{a_n,p,r\}_{(G)}=\biggl\{u(x)\in C^\infty(G):\sum_{n=0}^\infty a_n\|D^nu\| _r^p<\infty\biggr\}, $$
where $G$ can be a closed interval, the line, a ray, or the circle. The imbedding conditions depend on the form of the domain.
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