Firstly, we will establish the continuity of the addition and scalar multiplication maps for a sequential topology to get a $$ \mathbb K $$ -vector sequential topology and we will show the stability of the local $$ \mathbb K $$ -convexity by going to the sequential topology. Secondly, we will characterize the sequential closure of a kernel of a linear form and prove the geometric Hahn-Banach theorem in a $$ \mathbb K $$ -vector non-Archimedean sequential topology.

中文翻译：

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在非阿基米德值域上的 $$ \mathbb K $$-Vector Sequential Topology

首先，我们将建立序列拓扑的加法和标量乘法映射的连续性，以获得 $$ \mathbb K $$ -vector 序列拓扑，我们将展示局部 $$ \mathbb K $$ -convexity 的稳定性通过进入顺序拓扑。其次，我们将刻画线性形式核的顺序闭包，并在 $$ \mathbb K $$ -vector 非阿基米德顺序拓扑中证明几何 Hahn-Banach 定理。