Proposition (Criterion for being a subspace)

A subset $U\subset V$ is a Subspace of $V$ if

1. $0\in U$
2. $U$ is closed under addition and scalar multiplication.

Proposition (Sum of subspaces is a subspace)

If $U_1,U_2\subset V$ are subspaces of $V$, then $U_1+U_2$ is a subspace of V

Proposition ($U_1+U_2$ is the smallest subspace containing $U_1\cup U_2$)

If $U_1, U_2, W\subset V$ are subspaces such that $U_1\cup U_2\subset W$, then $U_1+U_2\subset W$.

Proposition (The span of vectors is a subspace)

If $V$ is a vector space, and $v_1,...,v_k\in V$, then $span(v_1,...,v_k)$ is a subspace of $V$, and is the smallest subspace containing the vectors $v_1,..,v_k$.