---
tag: 822WQ7S91T
kind: statement
role: theorem
name: Kolmogorov's normability criterion
msc: [46A03]
uses: [0S1FJ8VEX9, 02AS45T3H9, 3BAJ8ZTVPX, 0ZCY0Z80B6, 0V2WP9B1MF, PAX9SHDHDW]
about: [0V2WP9B1MF]
sources:
  - url: https://en.wikipedia.org/wiki/Kolmogorov's_normability_criterion
same_as:
  wikidata: Q25402330
  wikipedia: Kolmogorov's normability criterion
---

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[Hausdorff|0S1FJ8VEX9]] topological vector space over $\mathbb{K}$.
Then some [[norm|0V2WP9B1MF]] on $E$ induces the topology of $E$ if and only if $0$ has a [[convex|02AS45T3H9]] [[von Neumann bounded|3BAJ8ZTVPX]] [[neighbourhood|0ZCY0Z80B6]] in $E$.

## Remarks

If the topology comes from a norm, the open unit ball is a convex bounded neighbourhood of $0$.
Conversely, a convex bounded neighbourhood of $0$ contains a convex balanced open neighbourhood $U$ of $0$, which is still bounded.
Its [[Minkowski functional|PAX9SHDHDW]] $p_U$ is a seminorm, and boundedness of $U$ shows that the sets $t U$, $t > 0$, form a neighbourhood base at $0$, so $p_U$ induces the topology.
The Hausdorff property then forces $p_U(x) > 0$ for $x \ne 0$, so $p_U$ is a norm.
Example: the Fréchet space $C(\R)$ with the seminorms $p_n(f) = \sup_{\abs{t} \le n} \abs{f(t)}$ is not normable, because no neighbourhood of $0$ is bounded.
The space $L^p([0, 1])$ with $0 < p < 1$ is not normable either: it has bounded neighbourhoods of $0$ but no convex ones other than the whole space.
The Hausdorff hypothesis is needed: $\R$ with the indiscrete topology is a topological vector space.
In it, $\R$ itself is a convex bounded neighbourhood of $0$, but no norm induces this topology.

## Explanations

### L1 human

The theorem tells exactly when the topology of a topological vector space comes from a single norm.
Two features are needed near the origin: some neighbourhood must be convex, and it must also be small, in the sense of being bounded.
A norm ball has both features, and the theorem says that a set with both features can always be turned into a norm ball.

### L2 human

Local convexity means that the topology is generated by seminorms.
So the criterion says that a locally convex space is normable precisely when one seminorm suffices, up to equivalence.
It shows that spaces such as $C^\infty(\Omega)$, $H(\Omega)$ or the Schwartz space are not normable.
In each of them a bounded neighbourhood of $0$ would be relatively compact, which happens only in finite dimension.
A topological vector space with a bounded neighbourhood of $0$ is called locally bounded, so the criterion reads: normable means Hausdorff, locally convex and locally bounded.
The construction of the norm is the same Minkowski functional argument that links convex sets and seminorms in the theory of [[locally convex spaces|B6GY2KJMWW]].