Linear mapping/Normed spaces/Continuity/Characterization/Fact

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

${\displaystyle \varphi \colon V\longrightarrow W}$

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