---
tag: NPR76WM5DQ
kind: definition
name: Topology generated by a family of seminorms
msc: [46A03]
defines: ZVFXQ7S6FY
uses: [FP5GVCN8GV, 05EGHEQZAW, 0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $V$ be a [[vector space|05EGHEQZAW]] over $\mathbb{K}$, and let $\mathcal{P}$ be a family of [[seminorms|FP5GVCN8GV]] on $V$.
For $x \in V$, finitely many $p_1, \ldots, p_n \in \mathcal{P}$ and $\varepsilon > 0$, put
$$B(x; p_1, \ldots, p_n; \varepsilon) = \set{y \in V : p_i(y - x) < \varepsilon \text{ for } i = 1, \ldots, n}.$$
The *topology generated by* $\mathcal{P}$ is the [[topology|0CQHFPGJ45]] on $V$ defined as follows.
A set $U \subseteq V$ is open if for every $x \in U$ there are $p_1, \ldots, p_n \in \mathcal{P}$ and $\varepsilon > 0$ with $B(x; p_1, \ldots, p_n; \varepsilon) \subseteq U$.

## Remarks

With this topology $V$ is a topological vector space, each $p \in \mathcal{P}$ is continuous, and it is the coarsest topology on $V$ with these two properties.
A net $(x_i)$ converges to $x$ if and only if $p(x_i - x) \to 0$ for every $p \in \mathcal{P}$.
The topology is Hausdorff if and only if $\mathcal{P}$ separates points: for each $x \ne 0$ some $p \in \mathcal{P}$ has $p(x) \ne 0$.
Example: on $\R^X$, the space of all real functions on a set $X$, the seminorms $p_x(f) = \abs{f(x)}$ for $x \in X$ generate the topology of pointwise convergence, which is the product topology.
The check: a basic set $B(f; p_{x_1}, \ldots, p_{x_n}; \varepsilon)$ restricts the values at $x_1, \ldots, x_n$ only, as a basic open set of the product topology does.
Non-example: for $0 < p < 1$, the usual metric topology of $L^p([0, 1])$ is not generated by any family of seminorms.
Indeed, its only continuous linear form is $0$.
A nonzero continuous seminorm would give a nonzero continuous linear form by the Hahn–Banach theorem.
The topologies generated by families of seminorms are exactly the topologies of [[locally convex spaces|B6GY2KJMWW]].