Large and Small Subspaces of Hilbert Space
نویسندگان
چکیده
For example, Theorem 3 says that if V is a closed subspace of f2 and if V CQp for some p < 2, then V is finite-dimensional . On the other hand, the corollary to Theorem 4 states that there exist infinite-dimensional subspaces V of f 2 none of whose nonzero elements belongs to any f p -space (p < 2) . [For L2(0, 1) the results are somewhat different: (1) if V is a closed subspace of L 2(0, 1) and if V c L,, then V is finite-dimensional . Theorem 6 gives a condition for the finite-dimensionality of V in terms of Orlicz spaces, and by Theorem 5 this condition is best possible; in particular, L . cannot be replaced by Lq for any q < °O . (2) There exist infinite-dimensional subspaces of Lz none of whose nonzero elements is in any L q space (q > 2) (Theorem 7) ] .
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