Definition 1

Let E be a left vector space over $\mathbf{K}$. A mapping $p$ of E in $\mathbf{R}_+ = \{ 0, + \infty \}$, is called a semi-norm on E if it satisfies the following axioms : (SN_I) If $x \in \mathrm{E}$ and $\lambda \in \mathbf{K}$ then $p(\lambda x) = |\lambda| \, p(x)$. (SN_II) If $x \in \mathrm{E}$ and $y \in \mathrm{E}$ then $p(x + y) \leq p(x) + p(y)$.

Since $p(x) \leq p(y) + p(x - y)$ and $p(y) \leq p(x) + p(y - x)$, from $p(y - x) = p(x - y)$, we deduce