Weak topology

In mathematics, 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. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.

History[]

Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of functional analysis did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable.^[1] In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence.^[1] The weak topology is also called topologie faible and schwache Topologie.

The weak and strong topologies[]

Let ${\displaystyle \mathbb {K} }$ be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications ${\displaystyle \mathbb {K} }$ will be either the field of complex numbers or the field of real numbers with the familiar topologies.

Weak topology with respect to a pairing[]

Both the weak topology and the weak* topology are special cases of a more general construction for pairings, which we now describe. The benefit of this more general construction is that any definition or result proved for it applies to both the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs. This is also the reason why the weak* topology is also frequently referred to as the "weak topology"; because it is just an instance of the weak topology in the setting of this more general construction.

Suppose (X, Y, b) is a pairing of vector spaces over a topological field ${\displaystyle \mathbb {K} }$ (i.e. X and Y are vector spaces over ${\displaystyle \mathbb {K} }$ and b : X × Y → ${\displaystyle \mathbb {K} }$ is a bilinear map).

Notation. For all x ∈ X, let b(x, •) : Y → ${\displaystyle \mathbb {K} }$ denote the linear functional on Y defined by y ↦ b(x, y). Similarly, for all y ∈ Y, let b(•, y) : X → ${\displaystyle \mathbb {K} }$ be defined by x ↦ b(x, y).

Definition. The weak topology on X induced by Y (and b) is the weakest topology on X, denoted by