Let V be a K -Vectorspace, and let B 1 , B 2 , . . . be subsets of V such that B 1 ⊇ B 2 ⊇ . . . and for all i ∈ N ∖ { 0 } is ⟨ B i ⟩ = V. Prove or disprove that ⟨ ∩ i ∈ N B i ⟩ = V. The hint from our professor was that this is not true. Let's choose B k such that it is a subset of all other B i and also the smallest set. So every vector of B k is also in every other B i . Therefore, the intersection of all B i would be B k and we know that ⟨ B k ⟩ = V which would make the statement true. Where am I wrong? Any solution or hint would be highly appreciated.

Question

Let V be a       K  -Vectorspace, and let       B    1    ,      B    2    ,  .  .  . be subsets of V such that       B    1    ⊇      B    2    ⊇  .  .  . and for all   i  ∈      N    ∖  {  0  } is   ⟨      B    i    ⟩  =  V. Prove or disprove that   ⟨      ∩          i      ∈              N                  B    i    ⟩  =  V. The hint from our professor was that this is not true. Let&#039;s choose       B    k   such that it is a subset of all other       B    i   and also the smallest set. So every vector of       B    k   is also in every other       B    i  . Therefore, the intersection of all       B    i   would be       B    k   and we know that   ⟨      B    k    ⟩  =  V which would make the statement true. Where am I wrong? Any solution or hint would be highly appreciated.

akademiks1989rz · Accepted Answer

Nobody said that we have only finitely many       B    i  &#039;s. Suppose that   V  =      K   and that      B    n    =      {    x    ∈          K        ∣          |        x          |        &amp;lt;          1      n        }    .Then   (  ∀  n  ∈      N    )  :  ⟨      B    n    ⟩  =      K  . However,       ⋂          n      ∈              N                  B    n    =  {  0  }, and so      ⟨          ⋂              n        ∈                  N                            B      n        ⟩    =  {  0  }  .

Let V be a KK-Vectorspace, and let B_1,B_2,... be subsets of V such that B_1 supe B_2 supe ... and for all i in NN ∖{0} is ⟨B_i⟩=V. Prove or disprove that ⟨nn_(i in NN) B_i⟩=V.

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