Let V {\displaystyle {}V} be a K {\displaystyle {}K} -vector space with its dual space V ∗ {\displaystyle {}{V}^{*}} . Show that the orthogonal space U ⊥ ⊆ V ∗ {\displaystyle {}U^{\perp }\subseteq {V}^{*}} to a linear subspace U ⊆ V {\displaystyle {}U\subseteq V} , and the orthogonal space F ⊥ ⊆ V {\displaystyle {}F^{\perp }\subseteq V} to a linear subspace F ⊆ V ∗ {\displaystyle {}F\subseteq {V}^{*}} , are indeed linear subspaces.
