Annihilator dual space

Question

I am sitting on a problem. I know that $V$ is a $F$ Vector space and $U$ is a subspace of $V$. The set $U_0 := \{ f \in V^∗ ~|~ f(u)=0 \; \forall u \in U \}$ is called the Annihilator of $U$. I should now show that $U_0$ is a subspace of the dual space $V^∗$. My idea was very simple. Since we know that $U$ is subspace of $V$ we can take two elements $u_1$ and $u_2$ of $U$ and follow by definition of the Annihilator $u_1 + u_2 \Rightarrow f(u_1+\lambda u_2)=f(u_1)+f(\lambda u_2)$ which is a Subdualspace of $f(v)=0$. Is this okay?