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Can you give me an example that the following inclusion is strict $(U\cap U_1)+(U\cap U_2)\subseteq U\cap(U_1+U_2)$ if $U$ is a vector space and $U_1,U_2$ are subspaces of it.

Both are not necessarily subspaces, but is there a case s.t. RHS is one and LHS is not ?

ketum
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  • I assume you rather mean that $U,U_1,U_2$ are subspaces of a common vector space ($V$, say)? – Hagen von Eitzen Oct 21 '15 at 11:58
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    @Hagen von Eitzen I think you can German http://i.stack.imgur.com/6qlJy.png – ketum Oct 21 '15 at 12:03
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    Well, it seems the original problem statement is exactly as you translated it - which means that TZakrevskiy's answer is spot on. (It is an awful problem statement though and I still consider it quite likely that the author did not intend it this way) – Hagen von Eitzen Oct 21 '15 at 19:59

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If $U$ is a vector space and $U_i$ are its subspaces, then $U_1+U_2$ is also a subspace of $U$. Moreover, $U\cap U_i = U_i$ and $U\cap (U_1+U_2) = U_1+U_2$.

Hence your inclusion essentially writes

$$U_1+U_2\subseteq U_1+U_2,$$ which is always an equality.

TZakrevskiy
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  • Yes I think the equalities in the $2^{\text{nd}}$ line should be correct, but $U_1+U_2$ is not always a subspace right ? otherwise this post http://math.stackexchange.com/questions/71872/union-of-two-vector-subspaces-not-a-subspace is false – ketum Oct 21 '15 at 12:09
  • @ketum If $U_i$ are vector spaces, then their sum $U_1+U_2$ is always a vector space (easy to check by definition). However, their union $U_1\cup U_2$ is not necessarily a vector space. – TZakrevskiy Oct 21 '15 at 12:34