We define and study the free topological vector space $\mathbb{V}(X)$ over a Tychonoff space $X$. We prove that $\mathbb{V}(X)$ is a $k_ω$-space if and only if $X$ is a $k_ω$-space. If $X$ is infinite, then $\mathbb{V}(X)$ contains a closed vector subspace which is topologically isomorphic to $\mathbb{V}(\mathbb{N})$. It is proved that if $X$ is a $k$-space, then $\mathbb{V}(X)$ is locally convex if and only if $X$ is discrete and countable. If $X$ is a metrizable space it is shown that: (1) $\mathbb{V}(X)$ has countable tightness if and only if $X$ is separable, and (2) $\mathbb{V}(X)$ is a $k$-space if and only if $X$ is locally compact and separable. It is proved that $\mathbb{V}(X)$ is a barrelled topological vector space if and only if $X$ is discrete. This result is applied to free locally convex spaces $L(X)$ over a Tychonoff space $X$ by showing that: (1) $L(X)$ is quasibarrelled if and only if $L(X)$ is barrelled if and only if $X$ is discrete, and (2) $L(X)$ is a Baire space if and only if $X$ is finite.
给定一个拓扑群$G$，它可以作为一个拓扑子群嵌入到某个拓扑向量空间中（在实数域上），我们称$G$有不变线性跨距，如果$G$在任意嵌入到拓扑向量空间中的所有线性跨距同构为拓扑向量空间。对于任意集合$A$，令$\mathbb{Z}^{(A)}$是$|A|$的直和，$|A|$是具有吉洪诺夫积拓扑的离散整数群的许多副本。证明了拓扑群$\mathbb{Z}^{(A)}$具有不变的线性跨距.这回答了D的一个问题。Dikranjan et al.在积极的。证明了给定非离散序列空间$X$，$X$上的自由交换拓扑群$A(X)$是嵌入拓扑向量空间但不具有不变线性跨距的拓扑群的一个例子.
Metrizability of Topological Vector Space Valued Cone Metric Spaces
标量化方法是研究向量优化问题的一个重要工具，向量最优化问题的解可以通过求解标量最优化问题得到。Du（2010）[A note on cone metric fixing point theory and its equivalence，Nonlinear Analysis 72 2259-2261]应用这一方法研究了广义锥度量空间中压缩映射不动点定理的向量形式与一般意义下度量空间中不动点定理的标量形式的等价性。本文证明了由拓扑向量空间值锥度量诱导的拓扑与由非线性标量化函数得到的度量诱导的拓扑是一致的，即任何拓扑向量空间值锥度量空间都是可度量化的.
