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In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest or coarsest) topology on the set which makes all the functions continuous. For example, the product topology on a Cartesian product is defined to be the weak topology with respect to the projection maps of the product.

An example of a weak topology that is particularly important in functional analysis 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 fewest 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 convexIn mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not topological vector spaceIn mathematics, a topological vector space ''X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are.

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 '.

The weak* topology

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

defined by

remains continuous.

An important fact about the weak* topology is the Banach-Alaoglu theoremFunctional analysis The Banach-Alaoglu theorem (also known as Alaoglu's Theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak topology. A common proof identifies the unit ball with the weak topology as: the unit ball in X ' is compactIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e in the weak* topology.

Furthermore, the unit ball of X is compact in the weak topology if and only if X is reflexiveFunctional analysis This page concerns the reflexivity of a Banach space. For Paul Halmos' notion of the reflexivity of an operator algebra or a subspace lattice, see reflexive operator algebra. In functional analysis, a Banach space is called reflexive i.

See also:

• Weak-star operator topologyOperator theory In functional analysis, the weak- topology on the set B(H of bounded operators on a Hilbert space is the weak- topology obtained from the predual of B(H the trace class operators on H''. See also: Topologies on the set of operators on a Hi