Geodesic
Limits of Banach spaces. Imbedding theorems. Applications to Sobolev spaces of infinite order
Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 395-405 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $p>1$. Bibliography: 8 titles. 
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     author = {Yu. A. Dubinskii},
     title = {Limits of {Banach} spaces. {Imbedding} theorems. {Applications} to {Sobolev} spaces of infinite order},
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     year = {1981},
     volume = {38},
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     url = {http://geodesic.mathdoc.fr/item/SM_1981_38_3_a4/}
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Yu. A. Dubinskii. Limits of Banach spaces. Imbedding theorems. Applications to Sobolev spaces of infinite order. Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 395-405. http://geodesic.mathdoc.fr/item/SM_1981_38_3_a4/

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