Let V be a vector space and assume that T, U, W are sub spaces of V. Show that if T cup U W is a sub space of V, then two of these subspaces must be contained in the other one?

Answer: T⊂U⊂W are subspaces of V

Step-by-step explanation:

Proof: This is the easier direction.

If T⊂U⊂W or W⊂U⊂T then we have U⊂T⊂W = T or T⊂U⊂W = U

orT⊂U⊂W=W respectively.

SoT⊂U⊂W is a subspace as T, U and W are subspaces.

1st case : T⊂U⊂W is true Then the disjunction W⊂U⊂T or U⊂T⊂W is trivially true.

Let x∈W1 and y∈W2-W1.

By the definition of the union, we have x∈W∪T∪C and y∈T⊂U⊂W

As T∪U∪W is a subspace, x+y∈T∪C∪W which, again by the definition of the union, means that x+y∈W∪T∪C

V∈W∪T∪C

As V was arbitrary, as desired.