Necessary and sufficient condition: continuous map on semi-normed vector spaces

Question

Is there any set of necessary and sufficient conditions for a linear application between semi-normed vectorial spaces being continuous?

Answer 1

The situation here is similar to that in normed spaces. A linear map $f:$ between seminormed vector spaces $X\to Y$ (over $\Bbb K$, where $\Bbb K=\Bbb R$ or $\Bbb K=\Bbb C$) is is continuous iff it is continuous at the zero. Indeed, the necessity is trivial. So we shal prove only the sufficiency. Let $x\in X$ be an arbitrary element and $\varepsilon>0$ be an arbitrary real number. Since the map $f$ is continuous at the zero, there exists a real number $\delta>0$ such that $\|f(z)\|<\varepsilon$ for each $z\in X$ with $\|z\|<\delta$. Let $y\in X$ and $\|y-x\|<\delta$. Then $\|f(y)-f(x)\|= \|f(y-x)\|<\varepsilon$.