Let K {\displaystyle {}K} denote a field, let V {\displaystyle {}V} be a K {\displaystyle {}K} -vector space, and U ⊆ V {\displaystyle {}U\subseteq V} a linear subspace. Let v ∈ V {\displaystyle {}v\in V} with v ∉ U {\displaystyle {}v\notin U} . Show that there exists a linear form φ : V → K {\displaystyle {}\varphi \colon V\rightarrow K} satisfying φ ( U ) = 0 {\displaystyle {}\varphi (U)=0} and φ ( v ) = 1 {\displaystyle {}\varphi (v)=1} .
