The weak* topology on dual spaces is a crucial concept in functional analysis. It's defined on the dual space X* using seminorms and allows for a more relaxed notion of convergence compared to the norm topology.

Weak* topology is coarser than the weak topology on X*, making it easier to work with in certain contexts. The Banach-Alaoglu theorem, which states that the closed unit ball in X* is weak* compact, is a key result in this area.

The Weak Topology on Dual Spaces

Weak topology in dual spaces

• Defined on the dual space $X^$ of a normed space $X$ using the family of seminorms ${p_x : x \in X}$
  • Each seminorm $p_x(f)$ evaluates the absolute value of the functional $f$ at the point $x$, i.e., $p_x(f) = |f(x)|$
• Subbase for the weak* topology consists of sets ${f \in X^* : |f(x) - f_0(x)| < \varepsilon}$
  • Determined by a point $x \in X$, a functional $f_0 \in X^$, and a positive real number $\varepsilon > 0$
• Convergence in the weak topology is equivalent to pointwise convergence on $X$
  • A net $(f_\alpha)$ in $X^$ converges to $f \in X^$ in the weak topology if and only if $f_\alpha(x) \to f(x)$ for each $x \in X$

Weak vs weak topology comparison

• Weak topology on $X^$ defined by seminorms ${p_x : x \in X}$ with $p_x(f) = |f| |x|$
• Every subbase element of the weak topology is also a subbase element of the weak topology
  • For $x \in X$, $f_0 \in X^$, $\varepsilon > 0$, the set ${f \in X^ : |f(x) - f_0(x)| < \varepsilon}$ is open in the weak topology
    • Follows from the inequality $|f(x) - f_0(x)| \leq |f - f_0| |x|$
• The weak* topology is coarser than the weak topology on $X^*$
  • Every weak open set is also weakly open, but not conversely

Weak Closed and Compact Sets in Dual Spaces

Characteristics of weak sets

• A subset $A \subset X^$ is weak closed if and only if it is closed under pointwise limits
  • Equivalent to: for every net $(f_\alpha)$ in $A$ that converges to $f \in X^$ in the weak topology, $f \in A$
• Banach-Alaoglu theorem: the closed unit ball $B_{X^} = {f \in X^ : |f| \leq 1}$ is compact in the weak topology
• A subset $A \subset X^$ is weak compact if and only if it is weak closed and bounded in the norm topology
• Weak* compact sets in $X^*$ have the following properties:
  • Every net in $A$ has a subnet that converges in the weak topology to an element of $A$
  • $A$ is compact in the weak* topology if and only if every continuous linear functional on $(X^*, \text{weak})$ attains its maximum on $A$

Relationship of weak and weak topologies

• If $X$ is reflexive (canonical embedding $J : X \to X^{}$ is surjective), the weak* and weak topologies on $X^*$ coincide
  • In this case, weak closed (resp. compact) sets are the same as weakly closed (resp. compact) sets
• If $X$ is not reflexive, the weak* topology is strictly coarser than the weak topology on $X^*$
  • There exist subsets of $X^$ that are:
    • Weak closed but not weakly closed
    • Weak compact but not weakly compact