View source for Semi-norm

A finite non-negative function $p$ on a [[Vector space|vector space]] $E$ (over the field of real or complex numbers) satisfying the following conditions:
\begin{equation}
p(\lambda x) = |\lambda|p(x),\quad p(x+y)\leq p(x) + p(y)
\end{equation}
for all $x,y\in E$ and all scalars $\lambda$. An example of a semi-norm is a [[Norm|norm]]; the difference is that a semi-norm may have $p(x)$ with $x\ne 0$. If a semi-norm $p$ is defined on a vector space and if $f$ is a [[Linear functional|linear functional]] on a subspace obeying the condition $|f(x)|\leq p(x)$, then this functional can be extended to the entire space so that the extension satisfies the same condition (the [[Hahn–Banach_theorem | Hahn–Banach theorem]]). In mathematical analysis one frequently encounters separable topological vector spaces (cf. [[Topological vector space|Topological vector space]]) in which an 0-neighbourhood basis exists whose elements are convex sets. Such spaces are said to be locally convex. An open convex 0-neighbourhood in a locally convex space is of the form $\{x : p(x)<1)\}$, where $p$ is a continuous semi-norm. Nevertheless, in mathematical analysis one also encounters topological vector spaces (including spaces with a metrizable topology) on which there exist no non-trivial semi-norms. The simplest examples of such spaces are $L_q(0,1)$, $0<q<1$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Topological vector spaces" , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,   "Functional analysis" , McGraw-Hill  (1979)</TD></TR></table>