Vector space/K/Norms equivalent/Continuity/Fact

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, and let ${}\Vert {-}\Vert _{1}$ and ${}\Vert {-}\Vert _{2}$ be norms on ${}V$. Then the following statements are equivalent.

1. The two norms are equivalent.
2. The identity

   $V\longrightarrow V$

   is continuous when ${}V$ is endowed with the first norm on the left, and the second norm on the right, and vice versa.

3. The ${}\Vert {-}\Vert _{1}$-ball is bounded with respect to the ${}\Vert {-}\Vert _{2}$-norm, and vice versa.
4. There exist real numbers ${}a,b$ such that

   ${}\Vert {v}\Vert _{2}\leq a\Vert {v}\Vert _{1}\,$

   and

   ${}\Vert {v}\Vert _{1}\leq b\Vert {v}\Vert _{2}\,$

   holds for all ${}v\in V$.

Proof, Write another proof