We extend the well-known Gelfand–Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be b-Gelfand–Phillips if every limited set in E, which is bounded in the strong topology \(\beta (E,E')\) on E, is precompact in \(\beta (E,E').\) Several characterizations of b-Gelfand–Phillips spaces are given. The problem of preservation of the b-Gelfand–Phillips property by standard operations over locally convex spaces is considered. Also we explore the b-Gelfand–Phillips property in spaces C(X) of continuous functions on a Tychonoff space X. If \(\tau\) and \({\mathcal{T}}\) are two locally convex topologies on C(X) such that \({\mathcal{T}}_p\subseteq \tau \subseteq {\mathcal{T}}\subseteq {\mathcal{T}}_k,\) where \({\mathcal{T}}_p\) is the topology of pointwise convergence and \({\mathcal{T}}_k\) is the compact-open topology on C(X), then the b-Gelfand–Phillips property of the function space \((C(X),\tau )\) implies the b-Gelfand–Phillips property of \((C(X),{\mathcal{T}}).\) If additionally X is metrizable, then the function space \(\big (C(X),{\mathcal{T}}\big )\) is b-Gelfand–Phillips.

我们将 Banach空间的著名 Gelfand-Phillips 性质扩展到局部凸空间，如果E中的每个有限集都以强拓扑\(\beta ( E上的E,E')\)在\(\beta (E,E')\)中是预紧的。给出了b -Gelfand-Phillips 空间的几个特征。考虑了通过局部凸空间上的标准运算来保存b -Gelfand-Phillips 性质的问题。我们还探索了空间C ( X) 吉洪诺夫空间X上的连续函数。如果\(\tau\)和\({\mathcal{T}}\)是C ( X )上的两个局部凸拓扑，使得\({\mathcal{T}}_p\subseteq \tau \subseteq {\mathcal {T}}\subseteq {\mathcal{T}}_k,\)其中\({\mathcal{T}}_p\)是逐点收敛的拓扑，\({\mathcal{T}}_k\)是C ( X )上的紧开拓扑，则函数空间\((C(X),\tau )\)的 b -Gelfand –Phillips 性质蕴涵\((C( X),{\mathcal{T}}).\)另外，如果X可度量，则函数空间\(\big (C(X),{\mathcal{T}}\big )\)为b -Gelfand–Phillips。