线性空间理论中一个不能推广的证明

A Proof of a Statement about Linear Space that Cannot be Generalized

设V1,V2是数域F上的线性空间V的真子空间,很容易证明V1∪V2真包含在V中[1].对于V中的m个真子空间V1,V2,…,Vm,易见m∪i=1V1也真包含在V中.但m=2时的证明方法不能推广到任意的m,否则,可以得到一个假命题.

Let V be a vector space over a number field F. If V1, V2,…, Vm are non-trivial subspaces of V,then the union mUi=1 V1 is proper subset of V. A proof of this statement in the case m = 2 cannot be generalized to the case m i=1 ≥3 because this proof also hold for a vector space over a finite field and the statement is no longer true for finite fields if m≥3.