Dual space/Orthogonal space/Linear subspace/Exercise

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