Linear mapping/Normed spaces/Continuity/Characterization/Fact

Let ${}V$ and ${}W$ be normed ${}{\mathbb {K} }$ -vector spaces, and let

\varphi \colon V\longrightarrow W

denote a linear mapping. Then the following properties are equivalent.

${}\varphi$ is continuous.
${}\varphi$ is continuous in the origin.
The set
${\left\{\varphi (v)\mid v\in V,\,\Vert {v}\Vert =1\right\}}$

is bounded.