Seminorm topology
= Seminorm topology
{title2=$\tau_{\mathcal P}$}
Given a nonempty family $\mathcal P$ of finite-valued <seminorms> on a real or complex <vector space> $E$, its seminorm topology has basic open zero <neighbourhoods>
$$
U(p_1,\ldots,p_m;\varepsilon_1,\ldots,\varepsilon_m)
=\bigcap_{i=1}^m\{x\in E:p_i(x)<\varepsilon_i\},
\qquad p_i\in\mathcal P,\quad\varepsilon_i>0.
$$
Their translates form a <basis of a topology>. Only finite <intersections> are required: an infinite <intersection> might impose a stronger <topology>. Equivalently, put $q(x)=\max_i p_i(x)/\varepsilon_i$; then $q$ is a <seminorm> and $U=\{q<1\}$. See https://www.mat.univie.ac.at/~gue/lehre/25AFA/AFA.pdf[Günther Hörmann, Advanced Functional Analysis, section 4, neighbourhood systems from seminorms] for the general construction.
Each defining <seminorm> is <continuous>, since $|p(x)-p(y)|\leq p(x-y)$. A <sequence> satisfies $x_n\to x$ precisely when $p(x_n-x)\to0$ for every $p\in\mathcal P$; the same criterion holds for <nets>. Finitely many measurements can each meet a prescribed tolerance after a common index.
The <seminorm> axioms also establish the required joint <continuity> of <vector addition> and <scalar multiplication>:
$$
p\bigl((x+y)-(x_0+y_0)\bigr)\leq p(x-x_0)+p(y-y_0),
\qquad
p(\lambda x-\lambda_0x_0)\leq |\lambda|p(x-x_0)+|\lambda-\lambda_0|p(x_0).
$$
For a basic <neighbourhood>, apply these estimates to its finitely many <seminorms>; near $\lambda_0$ the <absolute value> $|\lambda|$ is bounded. This gives a <locally convex space> even when some directions are invisible to every measurement.