Linear Subspaces Closed under Setwise Addition

Theorem
Let $V$ be a $K$-vector space.

Let $M, N$ be linear subspaces of $V$.

Then $L := M + N$ is also a linear subspace of $V$, where $+$ denotes setwise addition.

Proof
It needs to be demonstrated that $L$ is closed under $+$ and $\circ$.

So let $m_1 + n_1, m_2 + n_2 \in L$.

Then $\left({m_1 + n_1}\right) + \left({m_2 + n_2}\right) = \left({m_1 + m_2}\right) + \left({n_1 + n_2}\right) \in L$.

It follows that $L$ is closed under $+$.

Now let $\lambda \in K, m + n \in L$.

Then $\lambda \circ \left({m + n}\right) = \left({\lambda \circ m}\right) + \left({\lambda \circ n}\right) \in L$.

It follows that $L$ is closed under $\circ$.

Hence the result, by definition of linear subspace.