Weak topology on a topological vector space

Let $E$ be a topological vector space over $\mathbb K$ , with topological dual $E'$ . The weak topology $\sigma(E,E')$ is the weakest topology on $E$ making every $\varphi\in E'$ continuous. It is the topology generated by the seminorms

p_\varphi(x)=|\varphi(x)|,\qquad \varphi\in E'.

Consequently, a net $x_i$ converges weakly to $x$ exactly when $\varphi(x_i)\to\varphi(x)$ for every $\varphi\in E'$ . The weak topology is no finer than the original topology because the functionals in $E'$ were originally continuous.

Neighborhoods and separation

A basic weak neighborhood of $x$ restricts only finitely many functionals:

\{y\in E:|\varphi_j(y-x)|<\varepsilon_j,\ 1\leq j\leq m\}.

The weak topology is Hausdorff exactly when $E'$ separates points of $E$ . This separation holds for Hausdorff locally convex spaces, but can fail for general topological vector spaces whose continuous dual is too small.

Weak versus original convergence

Original-topology convergence implies weak convergence. The converse is usually false in infinite dimensions: weak convergence records every scalar observation $\varphi(x_i)$ , but not uniform control over all observations. For example, the standard orthonormal basis in $\ell^2$ converges weakly to $0$ while every term has norm $1$ .

Weak versus weak-star

On a Banach dual space $E'$ , the weak topology is $\sigma(E',E'')$ , whereas the weak-star topology is $\sigma(E',E)$ , using only evaluations by elements of the specified predual. They coincide when the canonical image of $E$ exhausts $E''$ , as for reflexive Banach spaces, but not in general. Keeping the two entries in $\sigma(\,\cdot\,,\,\cdot\,)$ visible prevents this common ambiguity.

References

Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Publisher record. Relevant: Chapter IV on dual pairs and weak topologies.
John B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990. Publisher record. Relevant: Chapter V on weak convergence and duality.

Time-dependent functions

A weakly continuous path applies this topology to continuity in time.