weak topology induced by the image of . In other words, it is the coarsest topology such that the maps Tx, defined by from to the base field or remain continuous. Weak-* convergence A net in is convergent to in the weak-* topology if it converges pointwise: for all . In particular, a sequence of converges to provided that for all . In this case, one writes as . Weak-* convergence is sometimes called the simple convergence or the pointwise convergence. Indeed, it coincides with the pointwise convergence of linear functionals. Properties If is a separable (i.e. has a countable dense subset) locally convex space and H is a norm-bounded subset of its continuous dual space, then H endowed with the weak* (subspace) topology is a metrizable topological space. However, for infinite-dimensional spaces, the metric cannot be translation-invariant. If is a separable metrizable locally convex space then the weak* topology on the continuous dual space of is separable. Properties on normed spaces By definition, the weak* topology is weaker than the weak topology on . An important fact about the weak* topology is the Banach–Alaoglu theorem: if is normed, then the closed unit ball in is weak*-compact (more generally, the polar in of a neighborhood of 0 in is weak*-compact). Moreover, the closed unit ball in a normed space is compact in the weak topology if and only if is reflexive. In more generality, let be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let be a normed topological vector space over , compatible with the absolute value in . Then in , the topological dual space of continuous -valued linear functionals on , all norm-closed balls are compact in the weak-* topology. If is a normed space, a version of the Heine-Borel theorem holds. In particular, a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded. This implies, in particular, that when is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded). Thus, even though norm-closed balls are compact, X* is not weak* locally compact. If is a normed space, then is separable if and only if the weak-* topology on the closed unit ball of is metrizable, in