Weak topology

The weak topology $\sigma (X,X^{\ast })$ on a normed space $X$ is the coarsest topology making every member of $X^{\ast }$ continuous. A net converges weakly exactly when every bounded linear functional converges on it.

The weak closure ${\overline{K}}^w$ of $K$ is its closure in the weak topology. Thus $x\in {\overline{K}}^w$ exactly when every weak neighbourhood of $x$ meets $K$.

A sequence $(x_n)$ is weakly null when $f(x_n)\to 0$ for every member $f$ of the continuous dual space. It may remain bounded away from zero in norm.

A sequence $(x_n)$ is weakly Cauchy when $(f(x_n))$ is a Cauchy sequence for every continuous linear functional $f$. Its consecutive differences form a weakly null sequence.

In an infinite-dimensional normed space, the weak closure of the unit sphere is the closed unit ball. Every basic weak neighbourhood imposes only finitely many linear conditions, whose common kernel contains a nonzero direction that can move any interior point onto the sphere.

The weak-star topology $\sigma (X^{\ast },X)$ is pointwise convergence of functionals on $X$. If $X$ is nonreflexive, it is strictly weaker than the weak topology $\sigma (X^{\ast },X^{\ast \ast })$ on $X^{\ast }$.

If $X$ is separable, the weak-star topology on $B_{X^{\ast }}$ is metrizable. For a dense sequence $(x_n)$ in $B_X$, a compatible metric is obtained by summing bounded multiples of $2^{-n}|f(x_n)-g(x_n)|$.

For a weak-star compact set $K\subseteq X^{\ast }$, its Szlenk derivation at scale $\epsilon >0$ isIt removes points having a relatively weak-star open neighbourhood of norm diameter at most $\epsilon$.

If $X^{\ast }$ is separable and $K$ is a nonempty weak-star closed subset of $B_{X^{\ast }}$, then $K_{\epsilon }^{\prime }$ is a proper subset of $K$ for every $\epsilon >0$. Metrizability produces sequences witnessing membership in the derivative, while a Baire-category argument applied to a universal weakly null sequence yields the strict inclusion.

A subset of a Banach space is weakly compact when it is compact in the weak topology; it is relatively weakly compact when its weak closure is weakly compact.

A set is weakly sequentially compact when every sequence in it has a subsequence that converges weakly to a point of the set.

For a subset of a Banach space, relative weak compactness is equivalent to every sequence having a weakly convergent subsequence. In particular, weak compactness and weak sequential compactness coincide.

The closed convex hull of a weakly compact subset of a Banach space is weakly compact.