Topological vector space

Builds on Vector space Topological space Product topology Continuous map

Definition

Let $\mathbb{K}$ be $\R$ or $\C$ with its usual topology. A topological vector space over $\mathbb{K}$ is a vector space $E$ over $\mathbb{K}$ together with a topology on $E$ such that:

the addition $E \times E \to E$ , $(x, y) \mapsto x + y$ , is continuous;
the scalar multiplication $\mathbb{K} \times E \to E$ , $(\lambda, x) \mapsto \lambda x$ , is continuous.

Both products carry the product topology.

Remarks

Example: $\R^n$ with the Euclidean topology. Addition and scalar multiplication are given by polynomials in the coordinates, so both are continuous. Non-example: $\R$ as a vector space over $\R$ with the discrete topology. Addition is continuous, but scalar multiplication fails clause 2. If it were continuous, then the map $\lambda \mapsto \lambda \cdot 1$ from $\mathbb{K}$ to $E$ would be continuous. Then the preimage $\set{1}$ of the open set $\set{1}$ would be open in $\mathbb{K}$ , which is false. Many authors also require the topology to be Hausdorff. The definition extends to vector spaces over any topological field.

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Tag	`9CJ0VSYZ81`
URI	`m2://9CJ0VSYZ81`
Kind	definition
Writing	W0 stub
MSC	46A03, 57N17
Defines	Topological vector space
Uses	Vector space, Topological space, Product topology, Continuous map
Changed	2 November 2026

Topological vector space

Definition

Remarks

Reading path

Further reading

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