发表机构
Ben-Gurion University of the Negev; M. V. Lomonosov Moscow State University（内盖夫本-古里安大学; 米哈伊尔·瓦西里耶维奇·罗蒙诺索夫莫斯科国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究拓扑向量空间中Fréchet--Urysohn及Baire子群和线性子空间的结构，证明其局部有限维性，并将结果推广至更广泛的$\kappa$-Fréchet--Urysohn子群族。
AI 中文摘要
对于赋范空间$E$，我们用$E_w$和$E'_{w^*}$分别表示赋予弱拓扑$w$和弱$^*$拓扑$w^*$的空间$E$及其对偶空间$E'$。众所周知，对于Banach空间$E$，$E_w$和$E'_{w^*}$均为Fréchet--Urysohn拓扑空间当且仅当$E$是有限维的。该结论对$E_w$和$E'_{w^*}$的所有线性子空间仍然成立。此外，对于任意赋范空间$E$，$E_w$的每个Fréchet--Urysohn子群都是局部有限维的；即它包含一个含于有限维线性子空间中的开子群。类似地，对于任意Banach空间$E$，$E'_{w^*}$的每个Fréchet--Urysohn子群都是局部有限维的。对于Tychonoff空间$X$上的自由局部凸空间$L(X)$、其弱版本$L_p(X)$以及自由拓扑向量空间$V(X)$，也得到了类似的结果。同样的结论对$E_w$、$E'_{w^*}$、$L(X)$、$L_p(X)$和$V(X)$的Baire子群和Baire线性子空间也成立。我们强调，即使对于可度量化子群，这些结果也是新的且非平凡的。事实上，所有结果都是在拓扑向量空间的更广泛的$\kappa$-Fréchet--Urysohn子群族中建立的。$\kappa$-Fréchet--Urysohn空间类是我们证明结果的自然框架。
英文摘要
For a normed space $E$, we denote by $E_w$ and $E'_{w^*}$ the space $E$ and its dual $E'$ endowed with the weak topology $w$ and the weak$^*$ topology $w^*$, respectively. It is well known that for a Banach space $E$, both spaces $E_w$ and $E'_{w^*}$ are Fréchet--Urysohn topological spaces if and only if $E$ is finite-dimensional. This statement remains true for all linear subspaces of $E_w$ and $E'_{w^*}$. Moreover, for any normed space $E$, every Fréchet--Urysohn subgroup of $E_w$ is locally finite-dimensional; that is, it has an open subgroup contained in a finite-dimensional linear subspace. Similarly, for any Banach space $E$, every Fréchet--Urysohn subgroup of $E'_{w^*}$ is locally finite-dimensional. Analogous results are obtained for the free locally convex space $L(X)$, its weak version $L_p(X)$, and the free topological vector space $V(X)$ over a Tychonoff space~$X$. The same conclusions hold for Baire subgroups and Baire linear subspaces of $E_w$, $E'_{w^*}$, $L(X)$, $L_p(X)$, and $V(X)$. We emphasize that even for metrizable subgroups, these results are new and nontrivial. In fact, all results are established for the much wider family of $κ$-Fréchet--Urysohn subgroups of topological vector spaces. The class of $κ$-Fréchet--Urysohn spaces is the natural framework in which the proofs of our results work.
