---
tag: 1DRW7JP962
kind: definition
name: Weak topology
msc: [46A20]
defines: P6KHVSTPZV
uses: [R68HNB5SVR, ZVFXQ7S6FY, 4A82S5FVRT]
sources:
  - url: https://en.wikipedia.org/wiki/Weak_topology
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(X, Y, \langle \cdot, \cdot \rangle)$ be a [[dual pair|R68HNB5SVR]] over $\mathbb{K}$.
The *weak topology* $\sigma(X, Y)$ on $X$ is the [[topology generated by the family of seminorms|ZVFXQ7S6FY]] $p_y(x) = \abs{\langle x, y \rangle}$, $y \in Y$.
Let $E$ be a Hausdorff locally convex space with [[continuous dual|4A82S5FVRT]] $E'$, paired by $\langle x, f \rangle = f(x)$.
The *weak topology* of $E$ is $\sigma(E, E')$, and the *weak-\* topology* of $E'$ is $\sigma(E', E)$.

## Remarks

$\sigma(X, Y)$ is the coarsest topology on $X$ for which every map $x \mapsto \langle x, y \rangle$, $y \in Y$, is continuous; it is a Hausdorff locally convex topology.
A net $(x_i)$ converges to $x$ in $\sigma(X, Y)$ if and only if $\langle x_i, y \rangle \to \langle x, y \rangle$ for every $y \in Y$.
The continuous linear forms on $(X, \sigma(X, Y))$ are exactly the maps $x \mapsto \langle x, y \rangle$ with $y \in Y$.
Example: in $\ell^2$, the unit vectors $e_n$ converge to $0$ in $\sigma(\ell^2, \ell^2)$, since $\langle e_n, y \rangle = y_n \to 0$ for each $y \in \ell^2$.
Non-example: on $\ell^2$, the norm topology makes every map $x \mapsto \langle x, y \rangle$ continuous.
But it is not the coarsest such topology, so it is not the weak topology: the sequence $(e_n)$ converges weakly but not in norm.