Answer
$W$ is a subspace of $C[-1,1]$.
Work Step by Step
Let $W$ be a subset of $V$ such that
$$W=\{f : f(-1)=0\}, \quad V=C[-1,1].$$
Assume that $u=f(x), v=g(x)\in W$ and $c\in R$. Now, we have
(a) $W$ contains the zero vector $0(x)=0$.
(b) $u+v=(f+g)x=f(x)+g(x)$. Since $(f+g)(-1)=f(-1)+g(-1)=0$, then $u+v\in W$.
(c) $cu=(cf)=cf(x)$. Since $(cf)(-1)=cf(-1)=0$, then $cu\in W$
Hence, $W$ is a subspace of $C[-1,1]$.
