Let $X=(X,\tau)$ be a topological vector space whose continuous dual $X^*$ separates points (i.e., is T2). The weak topology $\tau_w$ on $X$ is defined to be the coarsest/weakest topology (that is, the topology with the fewest open sets) under which each element of $X^*$ remains continuous on $X$.

Weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space.