I was reading through Harris's Algebraic Geometry book, and was slightly perplexed by the following paragraph:
"Note that the set of hyperplanes in a projective space $\mathbb{P}^{n}$ is again a projective space, called the dual projective space and denoted $\mathbb{P}^{n^{*}}$. Intrinsically, if $\mathbb{P}^{n} = \mathbb{P}V$ is the projective space associated to a vector space V, the dual projective space $\mathbb{P}^{n^{*}} = \mathbb{P}(V^{*})$ is the projective space associated to the dual space $V^{*}$. More generally, if $\Lambda \cong \mathbb{P^{k}} \subset \mathbb{P^{n}}$ is a $k$-dimensional linear subspace, the set of (k+1)-planes containing $\Lambda$ is a projective space $\mathbb{P^{n-k-1}}$, and the space of hyperplanes containing $\Lambda$ is the dual projective space $(\mathbb{P^{n-k-1}})^{*}$. Intrinsically, if $\mathbb{P^{n}} = \mathbb{P}V$ and $\Lambda = \mathbb{P}W$ for some $(k+1)$-dimensional subspace $W \subset V$, then the space of $(k+1)$-planes containing $\Lambda$ is the projective space $\mathbb{P}(V/W)$ associated to the quotient, and the set of hyperplanes containing $\Lambda$ is naturally the projectivization $\mathbb{P}((V/W)^{*}) = \mathbb{P}(Ann(W)) \subset \mathbb{P}(V^{*})$ of the annihilator $Ann(W) \subset V^{*}$ of $W$."
I understand that in general, $V^{*}$ is defined as the set of linear functionals $f \colon V \to R$ and it makes sense that hyperplanes, but I can't picture intuitively how the hyperplanes would be associated with the dual projective space. In addition, I was having trouble trying to understand geometrically how the last sentence works.