论文标题
具有不变线性跨度的拓扑组
Topological groups with invariant linear spans
论文作者
论文摘要
鉴于拓扑组$ g $可以将作为拓扑亚组嵌入到某些拓扑矢量空间中（在真实的领域）中，我们说，如果在拓扑矢量中的所有线性跨越$ g $的所有线性跨度，则$ g $具有不变的线性跨度。 对于任意设置$ a $ a $ \ mathbb {z}^{（a）} $是$ | a | $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - 经过托管产品拓扑的副本的副本。我们表明拓扑组$ \ mathbb {z}^{（a）} $具有不变的线性跨度。这回答了D. Dikranjan等人的问题。积极。 我们证明，鉴于非二零顺序空间$ x $，免费的Abelian拓扑组$ a（x）$ a $ x $是拓扑组的一个示例，该拓扑组嵌入了拓扑矢量空间，但没有不变的线性跨度。
Given a topological group $G$ that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that $G$ has invariant linear span if all linear spans of $G$ under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set $A$ let $\mathbb{Z}^{(A)}$ be the direct sum of $|A|$-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group $\mathbb{Z}^{(A)}$ has invariant linear span. This answers a question of D. Dikranjan et al. in positive. We prove that given a non-discrete sequential space $X$, the free abelian topological group $A(X)$ over $X$ is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.
