On the Ascoli property for locally convex spaces and topological groups
On the Ascoli property for locally convex spaces and topological groups
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局部凸空间和拓扑群的阿斯科利性质
DOI：
10.1016/j.topol.2017.07.005
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
S. Gabriyelyan
中科院分区：
文献类型：
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作者：
S. Gabriyelyan
我们刻画了Ascoli空间，证明了Tychonoff空间X是Ascoli当且仅当从X上的自由局部凸空间L（X）到Ck（Ck（X））的正则映射是局部凸空间的嵌入.证明了非平凡局部凸空间的不可数直和不是Ascoli。若c 0桶空间E是弱Ascoli空间，则E对某个集合Γ线性同构于R Γ的稠密子空间.因此，Fréchet空间E是弱Ascoli当且仅当E= RN，其中N≤ ω。如果X是μ-空间和k R-空间（例如，可度量化），则C k（X）是弱Ascoli当且仅当X是离散的。如果X是μ-空间，则X上所有正则Borel测度的紧支撑空间MC（X）在弱拓扑中是Ascoli的当且仅当X是有限的.可度量化桶形空间E的弱对偶空间是Ascoli当且仅当E是有限维的。
We characterize Ascoli spaces by showing that a Tychonoff space X is Ascoli iff the canonical map from the free locally convex space L (X) over X into C k (C k (X)) is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a c 0-barrelled space E is weakly Ascoli, then E is linearly isomorphic to a dense subspace of R Γ for some set Γ. Consequently, a Fréchet space E is weakly Ascoli iff E= R N for some N≤ ω. If X is a μ-space and a k R-space (for example, metrizable), then C k (X) is weakly Ascoli iff X is discrete. If X is a μ-space, then the space M c (X) of all regular Borel measures on X with compact support is Ascoli in the weak⁎ topology iff X is finite. The weak⁎ dual space of a metrizable barrelled space E is Ascoli iff E is finite-dimensional.