Stability of epsilon-isometric embeddings into Banach spaces of continuous functions
Stability of epsilon-isometric embeddings into Banach spaces of continuous functions
复制标题
连续函数 Banach 空间中 epsilon 等距嵌入的稳定性
DOI：
10.1016/j.jmaa.2017.04.039
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Liu Chunyan
中科院分区：
文献类型：
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作者：
Zhou Yu;Zhang Zihou;Liu Chunyan
设X是一个巴拿赫空间，T是紧Hausdorff空间，C (T)是由T上赋上范数的所有连续函数构成的实巴拿赫空间。我们表明,如果存在一个标准ε等距嵌入f: X→C (T),还有非空的闭子集S⊆T和线性等距嵌入g: X→跨度‾(f (X) | S)⊂C (S)定义为g (u) = lim n→∞f (2 n u) | S 2 n为每个u∈X满足为f (u) | S−g (u)为≤4ε为所有u∈X利用这个结果和众所周知的同时扩展运营商E: C (S)→C (T),我们也证明标准存在的ε等距嵌入f:X→C (T)表示存在线性等距嵌入E°g: X→span不要紧（E (f (X)| S)），只要T是可度量的。这些结论概括了几个众所周知的结果。对于任何紧化Hausdorff空间(参见。紧度量空间)T，我们进一步得到如果g (X)在span中（f (X)| S）（resp.）E°g (X)以span形式补齐（E (f (X)| S)），则标准ε-等距嵌入f: X→C (T)稳定。
Let X be a Banach space, T be a compact Hausdorff space and C (T) be the real Banach space of all continuous functions on T endowed with the supremum norm. We show that if there exists a standard ε-isometric embedding f: X→ C (T), then there are nonempty closed subset S⊆ T and a linearly isometric embedding g: X→ span‾(f (X)| S)⊂ C (S) defined as g (u)= lim n→∞ f (2 n u)| S 2 n for each u∈ X satisfying that‖ f (u)| S− g (u)‖≤ 4 ε for all u∈ X. Making use of this result and the well known simultaneous extension operator E: C (S)→ C (T), we also prove that the existence of a standard ε-isometric embedding f: X→ C (T) implies the existence of a linearly isometric embedding E∘ g: X→ span‾(E (f (X)| S))⊆ C (T) whenever T is metrizable. These conclusions generalize several well-known results. For any compact Hausdorff space (resp. compact metric space) T, we further obtain that if g (X) is complemented in span‾(f (X)| S)(resp. E∘ g (X) is complemented in span‾(E (f (X)| S))), then the standard ε-isometric embedding f: X→ C (T) is stable.