摘要
作者在[1]中讨论了域上有限维线性空间的可覆盖性。本文进一步研究对无限维线性空间的可覆盖性与其基域的基数之间的关系,并得到了有趣的若干结果。1 预备知识设V是域F上的线性空间,如果V中的任一元素都是V的子集S中元素的(有限)线性组合,则称S是V的生成元集。如果S中的的任何有限子集均是线性无关的。
definition: A class {V}_(i∈j) of finite dimension non-zero subspace V_i of a linear spacc V over a field F is called a cover of V if V=∪_V_I i∈j. The cover of V is called finite-cover(countable-cover or continuum-cover respectively) if |I|<∞(|I|)<|N| or |I|=|R|. In this paper the following results are obtained: Theorem 1. V has finite-cover iff |F|<∞ and dimV<∞. Theorem 2. If dimV=|N| then V has countable-cover iff |F|<∞ or |F|=|N|. Theorem 3. 1) If dimV=|N| then V has continuum-cover iff |F|=|R|. 2) If dimV=|R| then V has continuum-cover iff |F|<∞ or |F|=|N| or |F|=|R|
