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.

Karsyn Beltran

Karsyn Beltran

Answered 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'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.

Answer & Explanation



Beginner2022-07-18Added 16 answers

Nobody said that we have only finitely many B i 's. Suppose that V = K and that
B n = { x K | x | < 1 n } .
Then ( n N ) : B n = K . However, n N B n = { 0 }, and so
n N B n = { 0 } .

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