Limits of Banach spaces. Imbedding theorems. Applications to Sobolev spaces of infinite order

For a sequence of Banach spaces ${X_1}\supset{X_2}\supset\dotsb$, a concept of limit $X_\infty=\lim_{r\to\infty}X_r$ is introduced that is a natural generalization of the concept of the limit of a monotonically decreasing numerical sequence. Necessary and sufficient conditions are obtained for an imbedding $X_\infty\subset Y_\infty$ and for a compact imbedding. Applications are given to the Sobolev spaces of infinite order $W^\infty\{a_\alpha,p\}$.
Necessary and sufficient conditions bearing an algebraic character are established for the imbedding $W^\infty\{a_\alpha,2\}(\mathbf R^\nu)\subset W^\infty\{b_\alpha,2\}(\mathbf R^\nu)$. Sufficient algebraic imbedding conditions are obtained for the spaces $W^\infty\{a_\alpha,p\}(\mathbf R^1)$ for any $p>1$.
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