Euclidean space/Affine subspace/Perpendicular/Fact/Proof

Proof

We write $P_{1} - P_{2} = u_{1} + u_{2} + u$ with $u_{1} \in U_{1}$ , $u_{2} \in U_{2}$ , and $u \in {(U_{1} + U_{2})}^{⊥}$ ; such a decomposition does always exist, $u_{1}, u_{2}$ are not uniquely determined (in case $U_{1} \cap U_{2} \neq 0$ ), but $u$ is uniquely determined. We have

P_{1} - u_{1} = P_{2} + u_{2} + u,

and $Q_{1} := P_{1} - u_{1} \in E_{1}$ , and $Q_{2} := P_{2} + u_{2} \in E_{2}$ . The distance between $Q_{1}$ and $Q_{2}$ is $‖ u ‖$ . For arbitrary points $R_{1} = Q_{1} + v_{1} \in E_{1}$ and $R_{2} = Q_{2} + v_{2} \in E_{2}$ fulfilling $v_{1} \in U_{1}$ and $v_{2} \in U_{2}$ , we have

\begin{matrix} d (R_{1}, R_{2})^{2} & = ‖ R_{1} - R_{2} ‖^{2} \\ = ‖ v_{1} - v_{2} + u ‖^{2} \\ = ⟨ v_{1} - v_{2} + u, v_{1} - v_{2} + u ⟩ \\ = ⟨ v_{1} - v_{2}, v_{1} - v_{2} ⟩ + ⟨ u, u ⟩ \\ \geq ⟨ u, u ⟩, \end{matrix}

that is,

d (R_{1}, R_{2}) \geq ‖ u ‖ .

To fact