Linear subspace/Dual space/Orthogonal space/Correspondence/Fact

Let $V$ be a $K$-vector space with dual space $V^{\ast }$. Then the following statements hold.

1. For linear subspaces $U\subseteq U^{\prime }\subseteq V$, we have

   $U^{\perp }\supseteq U{\text{'}}^{\perp }.$

2. For linear subspaces $F\subseteq F^{\prime }\subseteq V^{\ast }$, we have

   $F^{\perp }\supseteq F{\text{'}}^{\perp }.$

3. Let $V$ be finite-dimensional. Then

   ${\left(U^{\perp }\right)}^{\perp }=U$

   and

   ${\left(F^{\perp }\right)}^{\perp }=F.$

4. Let $V$ be finite-dimensional. Then

   ${\dim }_K\left(U^{\perp }\right)={\dim }_K\left(V\right)-{\dim }_K\left(U\right)$

   and

   ${\dim }_K\left(F^{\perp }\right)={\dim }_K\left(V\right)-{\dim }_K\left(F\right).$

Proof, Write another proof