Annihilator in dual space

Question

Let U and W are subspaces of a vector space V. If U is subset of W, inh(W) is subset of inh(U). Is the Converse true? How?

Answer 1

No, not in general. The annihilator of a subset $U$ is the same as the annihilator of the linear span of $U$. So if two sets have the same linear span, they have the same annihilator, even though they are not equal.

Example: in the one-dimensional vector spae $V = \mathbb R$, the two subsets $U = \{5\}$ and $V = \{8\}$ both have annihilator $\{0\}$.

Answer 2

Let $u\in U$ and suppose it's not in $W$. Then extend a basis $(w_i)$ of $W$ by $u$ and possibly other $(v_j)$'s to obtain a basis of $V$.

Then define $f\in V^*$ so that $f(w_i) =0, \ f(u) =1, \ f(v_j) =0$.

This $f$ satisfies $f\in\mathrm{inh}(W) $, so $f\in\mathrm{inh}(U)$ by hypothesis, so $f(u)$ should be $0$, contradiction.