Understanding Weak and Weak* Topologies in Infinite Dimensional Normed Spaces

Branch of mathematical analysis concerned with infinite-dimensional topological vector spaces, often spaces of functions.

In the realm of infinite dimensional normed spaces, the concepts of weak and weak* topologies play a crucial role. These topologies provide a framework for understanding the convergence of sequences and nets, and they are instrumental in the study of dual spaces and reflexive spaces.

What are Weak and Weak* Topologies?

In the context of normed spaces, a topology is a mathematical structure that allows us to define concepts such as continuity, convergence, and compactness. The weak topology and the weak* topology are two such structures that we often encounter in infinite dimensional normed spaces.

The weak topology on a normed space is the coarsest topology that makes all continuous linear functionals continuous. In other words, a sequence (or net) converges weakly if and only if it converges pointwise under all continuous linear functionals.

The weak* topology, on the other hand, is a topology that we define on the dual space of a normed space. It is the coarsest topology that makes all evaluations at points of the original space continuous.

Relationship between Weak and Weak* Topologies and the Strong Topology

The strong topology, also known as the norm topology, is the topology that we usually consider on a normed space. It is defined by the norm of the space, and a sequence converges strongly if and only if it converges in norm.

The weak and weak* topologies are coarser than the strong topology, meaning they have fewer open sets. As a result, a sequence that converges weakly (or weak*ly) may not converge strongly. However, if a sequence converges strongly, it also converges weakly.

Applications of Weak and Weak* Topologies in Functional Analysis

The weak and weak* topologies have many applications in functional analysis. For instance, they are used in the study of dual spaces and reflexive spaces. They also play a crucial role in the Banach-Alaoglu theorem, which states that the closed unit ball in the dual space of a normed space is compact in the weak* topology.

Furthermore, weak and weak* convergences are used in the study of operators between infinite dimensional spaces. They are also used in the study of partial differential equations, where weak solutions are often considered.

In conclusion, the concepts of weak and weak* topologies are fundamental in the study of infinite dimensional normed spaces. They provide a framework for understanding the convergence of sequences and nets, and they have many applications in functional analysis.