---
tag: ZVFXQ7S6FY
kind: concept
name: Topology generated by a family of seminorms
msc: [46A03]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space
---

## Summary

A family of [[seminorms|FP5GVCN8GV]] on a vector space defines a topology.
In it, a net converges exactly when it converges for every seminorm of the family.
With this topology the vector space is a topological vector space, and in fact a [[locally convex space|B6GY2KJMWW]].
This is the standard way to put a topology on function spaces such as spaces of smooth functions, where no single norm is suitable.

## Explanations

### L1 human

Sometimes one way of measuring size is not enough, and one uses many measuring sticks at once.
Two vectors are declared close when they are close for finitely many of the sticks at a time.
For example, on the space of continuous functions on $\R$, the seminorms $p_n(f) = \sup_{\abs{t} \le n} \abs{f(t)}$ give the topology of uniform convergence on bounded intervals.
A sequence $f_k$ converges to $f$ in this topology when $p_n(f_k - f) \to 0$ for each $n$ separately.
The functions $f_k(t) = t / k$ converge to $0$ in this sense, although they do not converge uniformly on all of $\R$.