Lp(μ,X)中的性质(U)

PROPERTY (U) IN L p(μ,X)

证明了下面的两个结论：（１）Ｌｐ（μ，Ｙ）是Ｌｐ（μ，Ｘ）的Ｃｈｅｂｙｓｈｅｖ子空间的充要条件是Ｌｑ（μ，Ｙ）是Ｌｑ（μ，Ｘ）的Ｃｈｅｂｙｓｈｅｖ子空间（１≤ｐ，ｑ≤∞）；（２）Ｌｐ（μ，Ｙ）在Ｌｐ（μ，Ｘ）中具有性质（Ｕ）的充要条件是Ｌｑ（μ，Ｙ）在Ｌｑ（μ，Ｘ）中有性质（Ｕ）（１≤ｐ，ｑ＜∞）．并且证明：若Ｘ自反，ＹＸ为闭子空间，则Ｙ有性质（Ｕ）（或是Ｃｈｅｂｙｓｈｅｖ子空间）可得出Ｌ１（μ，Ｙ）在Ｌ１（μ，Ｘ）中有性质（Ｕ）（或是Ｃｈｅｂｙｓｈｅｖ子空间）

We get the following two main results:If Y is a closed subspace of Banach space X, then (1) L p(μ,Y) is the Chebyshev subspace of L p(μ,X) if and only if L q(μ,Y) is the Chebyshev subspace of L q(μ,X)(1≤ p,q≤∞.(2) L p(μ,Y) has property (U) in L p(μ,X) if and only if L q(μ,Y) has property (U) in L q(μ,X)(1≤p,q<∞) .And we also proved that if Y is a closed subspace of reflexive Banach space X and Y has property (U) (Chebyshev), then L p(μ,Y) has property (U) ( respectly, Chebyshev)in L p(μ,X) .