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## Homework Statement

Prove that the intersection of any collection of subspaces of V is a subspace of V.

## Homework Equations

To show that a set is a subspace of a vector space, I need to show that there exists an additive identity, and that the set is closed under addition and scalar multiplication.

For notation I will let F denote

**R**or

**C**.

## The Attempt at a Solution

Consider a collection of subspaces U_1, U_2, ..., U_m of a vector space V.

Clearly any intersection of these subspaces will contain 0, and hence the intersection contains the additive identity.

Now consider two elements, u and w, of any arbitrary intersection of U_1, U_2, ..., U_m. Since u and w are in the intersection of U_1, U_2, ..., U_m, u and w are elements of U_1 and U_2 and ... and U_m. Therefore their addition, u + w must be an element of U_1 and U_2 and ... and U_m. So u + w is an element of any intersection of U_1, U_2, ..., U_m and hence this intersection is closed under addition.

Now consider a scalar c in F and an element u of any arbitrary intersection of U_1, U_2, ..., U_m. Since u is in the intersection of U_1, U_2, ..., U_m, u is an element of U_1 and U_2 and ... and U_m. Therefore cu is an element of U_1 and U_2 and ... and U_m. So cu is an element of any intersection of U_1, U_2, ..., U_m and hence this intersection is closed under scalar multiplication.

Any feedback would be greatly appreciated. If it is sloppy, please let me know what I can do to improve my proof writing. Thanks again!