A particularly important example of a weak topology is that on a normed vector space with respect to its (continuous) dual. The remainder of this article will deal with this case.

Every normed vector space *X* is, by using the norm to measure distances, a metric space and hence a topological space. This topology on *X* is also called the **strong topology**. The **weak topology** on *X* is defined using the continuous dual space *X* '. This dual space consists of all linear functions from *X* into the base field **R** or **C** which are continuous with respect to the strong topology. The weak topology on *X* is the weakest topology (the topology with the least open sets) such that all elements of *X* ' remain continuous. Explicitly, a subset of *X* is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which being an intersection of finitely many sets of the form φ^{-1}(*U*) with φ in *X* ' and *U* an open subset of the base field **R** or **C**. A sequence (*x*_{n}) in *X* converges in the weak topology to the element *x* of *X* if and only if φ(*x*_{n}) converges to φ(*x*) for all φ in *X* '.

If *X* is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and *X* is a locally convex topological vector space.

The dual space *X* ' is itself a normed vector space by using the norm ||φ|| = sup_{||x||≤1}|φ(*x*)|. This norm gives rise to the **strong topology** on *X* '. One may also define a **weak* topology** on *X* ' by requiring that it be the weakest topology such that for every *x* in *X*, the substitution map

- Φ
_{x}:*X*' →**R**or**C**

- Φ
_{x}(φ) = φ(*x*)

An important fact about the weak* topology is the Banach-Alaoglu theorem: the unit ball in *X* ' is compact in the weak* topology.

Furthermore, the unit ball of *X* is compact in the weak topology if and only if *X* is reflexive.