Vector space/K/Inner product/Finite-dimensional/Orthogonal complement/Structural properties/Fact

Let $V$ be a $𝕂$-vector space, endowed with an inner product $⟨-,-⟩$. Then the following statements hold.

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

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

2. We have $0^{\perp }=V$ and $V^{\perp }=0$.
3. Let $V$ be finite-dimensional. Then we have

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

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

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

Write a proof