Prove dual space Y* is subspace of X* if X is subspace of Y

Question

We have V - vector space and U, W ⊆ V (subspaces). U* and W* are the dual spaces accordingly. I have to prove that if U is subspace of W, then W* is subspace of U*. I am honestly baffled because the concept of duality is still fresh in my mind. What I tried is proving that W* is closed around scalar multiplication and addition, but because it's a space consisting of functionals I don't know how to proceed. Any hints would be welcome.

Answer 1

Let $U\subset W$, and $f \in W^*$. So $$\forall w \in W, f(w) \in \mathbb{F}.$$ ($\mathbb{F}$ is the scalar field). As $U \subset W$, then $\forall u \in U$, $u \in W$. Then, $f(u) \in \mathbb{F}, \forall u \in U$. So, as $f$ is linear in $W$, then $f$ is linear in $U$. Therefore, $f \in U^*$.