Let $c∈\Bbb R$ and $U⊂\Bbb R^n$,$U \ne∅$ ($U$ is a nonempty subset). Further let $〈·,·〉:\Bbb R^n×\Bbb R^n→\Bbb R$ be the standard inner product. Define
$$U := \{ v ∈ \Bbb R^n : ∀ u ∈ U : 〈 v , u 〉 = c \} .$$
Show that $Uc$ is an affine subspace.
I think I should use the Inner standard product.
Edit, solution :
we have the Characterisation of affine subspaces
Let S ⊂ R^n, such that a,b ∈ S ,λ∈R ==> λa + (1−λ)b∈ S .Then S is already an affine subspace.
Solution :
we assume u1,u2 ∈ U and λ∈R so we have ⟨v,λu2 + (1−λ)u1⟩=c. ==> the property of Inner Product ⟨v,λu2⟩+⟨v,(1−λ)u1⟩ then we pull out lambda out λ⟨v,u2⟩+(1−λ)⟨v,u1⟩ ==> ⟨v,u2⟩ = c and ⟨v,u1⟩ = c ==> λc + (1−λ) c = λc + c - λc = c