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

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

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

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

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

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

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