The on dual spaces is a crucial concept in functional analysis. It's defined on the X* using seminorms and allows for a more relaxed notion of convergence compared to the .
Weak* topology is coarser than the weak topology on X*, making it easier to work with in certain contexts. The , which states that the closed unit ball in X* is , 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 {px:x∈X}
Each seminorm px(f) evaluates the absolute value of the functional f at the point x, i.e., px(f)=∣f(x)∣
Subbase for the weak* topology consists of sets {f∈X∗:∣f(x)−f0(x)∣<ε}
Determined by a point x∈X, a functional f0∈X∗, and a positive real number ε>0
Convergence in the weak* topology is equivalent to pointwise convergence on X
A net (fα) in X∗ converges to f∈X∗ in the weak* topology if and only if fα(x)→f(x) for each x∈X
Weak* vs weak topology comparison
Weak topology on X∗ defined by seminorms {px:x∈X} with px(f)=∥f∥∥x∥
Every subbase element of the weak* topology is also a subbase element of the weak topology
For x∈X, f0∈X∗, ε>0, the set {f∈X∗:∣f(x)−f0(x)∣<ε} is open in the weak topology
Follows from the inequality ∣f(x)−f0(x)∣≤∥f−f0∥∥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⊂X∗ is if and only if it is closed under pointwise limits
Equivalent to: for every net (fα) in A that converges to f∈X∗ in the weak* topology, f∈A
Banach-Alaoglu theorem: the closed unit ball BX∗={f∈X∗:∥f∥≤1} is compact in the weak* topology
A subset A⊂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∗,weak*) attains its maximum on A
Relationship of weak* and weak topologies
If X is reflexive (canonical embedding J:X→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
Key Terms to Review (28)
Banach-Alaoglu Theorem: The Banach-Alaoglu Theorem states that in a normed space, the closed unit ball in the dual space is compact in the weak* topology. This theorem connects the concepts of dual spaces, weak topologies, and compactness, which are fundamental in understanding properties of linear functionals and their applications.
Bidual space: A bidual space is the dual of the dual space of a vector space, denoted as $X^{**}$ for a vector space $X$. It consists of all continuous linear functionals defined on the dual space $X^*$, allowing a deeper understanding of the properties of the original space $X$. Bidual spaces provide insights into reflexivity, natural embeddings, and weak* topologies, revealing how these structures interact within functional analysis.
Bounded Linear Functional: A bounded linear functional is a linear map from a vector space to its underlying field that is continuous and has a finite norm. This means that there exists a constant such that the absolute value of the functional applied to any vector in the space is less than or equal to this constant times the norm of that vector. Bounded linear functionals play a crucial role in understanding dual spaces, the Hahn-Banach Theorem, and the weak* topology, as they help in extending functionals while preserving continuity and facilitate various analyses of convergence and topology in functional spaces.
Bounded linear functional: A bounded linear functional is a specific type of linear functional that is continuous and maps elements from a normed vector space to the underlying field, typically the real or complex numbers. This concept is essential for understanding dual spaces, as it relates directly to the behavior of linear functionals in relation to the norms of the spaces they operate on.
C(x): In the context of weak* topology on dual spaces, c(x) represents a continuous linear functional on a topological vector space X. This functional is defined based on the evaluation of elements in the dual space, emphasizing how elements of the space interact with linear functionals. The significance of c(x) lies in its ability to demonstrate convergence and continuity properties in the dual space, particularly under the weak* topology.
Continuous Linear Operator: A continuous linear operator is a mapping between two normed vector spaces that preserves linearity and is continuous with respect to the topology induced by the norms of those spaces. This concept is crucial as it links the properties of boundedness and continuity, providing foundational insights in functional analysis and allowing for the examination of operators in various settings such as dual spaces and compactness.
Dual Space: The dual space of a vector space consists of all linear functionals defined on that space. It captures the idea of measuring or evaluating vectors in terms of how they interact with linear functionals, which are themselves linear maps that take vectors as input and return scalars.
Duality pairing: Duality pairing is a concept that establishes a relationship between a vector space and its dual space, providing a way to evaluate functionals on vectors. This pairing allows us to capture how vectors and functionals interact, offering insights into both the structure of vector spaces and the behaviors of linear transformations. It is foundational in defining weak* topology, where the convergence of functionals in the dual space is analyzed relative to elements in the original space.
Fréchet: Fréchet refers to a type of topology used in functional analysis, particularly associated with the concept of convergence in spaces that may be infinite-dimensional. This topology generalizes notions of distance and convergence, making it essential for understanding the behavior of sequences and functionals in various spaces, such as dual spaces and compact operators.
Nets: Nets are generalized sequences that allow for the convergence of a wider class of functions and topological spaces than traditional sequences. They are particularly useful in functional analysis as they provide a way to study continuity and compactness in spaces that may not be first-countable, which is crucial for understanding weak* topology on dual spaces.
Nicolas Bourbaki: Nicolas Bourbaki is a collective pseudonym used by a group of primarily French mathematicians who, starting in the 1930s, sought to reformulate mathematics on an extremely abstract and formal basis. The group's work emphasized rigorous definitions, theorems, and proofs across various branches of mathematics, significantly impacting modern mathematical thought and education.
Norm Topology: Norm topology refers to the topology defined on a normed vector space, where the open sets are determined by the norms of the vectors in that space. This topology gives a framework for understanding convergence and continuity in the space, which becomes crucial when discussing concepts like weak and weak* topologies. In this setting, convergence is based on the norm, but other weaker notions of convergence arise when considering dual spaces and functionals.
Reflexivity: Reflexivity refers to a property of a linear functional where it corresponds to an element in a dual space such that a specific natural embedding exists between the original space and its dual, leading to the conclusion that the original space is isomorphic to its bidual. This concept is pivotal in understanding the relationship between spaces and their duals, as well as the nature of continuity and convergence in weak* topology.
Reflexivity: Reflexivity is a property of a Banach space that indicates it is naturally isomorphic to its double dual, meaning that every continuous linear functional on the dual space can be represented as evaluation at a point in the original space. This concept is crucial in understanding weak topologies, the duality of spaces, and how reflexive spaces maintain certain desirable properties in functional analysis.
Separability: Separability is a property of a topological space that indicates whether it contains a countable dense subset. In functional analysis, separability is crucial as it helps in understanding the structure of spaces, especially in relation to dual spaces and their weak* topologies. This concept reveals important features about the size and behavior of functionals within these spaces, highlighting connections between the original space and its dual or bidual counterparts.
Sequences in dual spaces: Sequences in dual spaces refer to ordered collections of continuous linear functionals defined on a Banach space, where the dual space consists of all such functionals. These sequences play a crucial role in understanding convergence concepts in the context of weak* topology, which is essential for studying functional analysis. In this framework, sequences can reveal properties about the compactness, boundedness, and continuity of mappings between various spaces.
Sequentially compact: A set is called sequentially compact if every sequence of points in the set has a subsequence that converges to a limit within the same set. This property is crucial for understanding the behavior of sequences and their limits, particularly in spaces where traditional compactness may not apply directly. Sequential compactness is closely related to various important concepts, such as continuity, convergence, and completeness in topological spaces.
Stephen Banach: Stephen Banach was a Polish mathematician known for his foundational work in functional analysis, particularly in the development of Banach spaces and the theory of linear operators. His contributions have had a profound impact on the field, establishing concepts that are essential to understanding dual spaces and weak* topology.
Strong Convergence: Strong convergence refers to a type of convergence in a normed space where a sequence of elements converges to a limit in the sense that the norm of the difference between the sequence and the limit approaches zero. This concept is crucial when dealing with bounded linear operators, as it ensures stability and continuity in various mathematical settings, including Banach spaces and Hilbert spaces.
Strong topology: Strong topology is a type of topology on a vector space that is generated by a family of seminorms, making it stronger than the weak topology. This means that a sequence converges in strong topology if it converges in norm, and it plays an important role in understanding dual spaces and reflexive spaces. In reflexive spaces, strong topology helps clarify the relationships between weak and strong convergence, while in dual spaces, it connects closely with weak* topology, highlighting how functionals behave under different convergence notions.
Strong vs. Weak Convergence: Strong convergence refers to a type of convergence where a sequence of elements in a normed space converges to a limit with respect to the norm, meaning that the distance between the sequence and the limit approaches zero. Weak convergence, on the other hand, involves convergence with respect to a weaker topology, where a sequence converges if it converges in terms of all continuous linear functionals applied to the elements of the sequence. Understanding these concepts is crucial when dealing with dual spaces and their topologies.
Weak Convergence: Weak convergence refers to a type of convergence in a topological vector space where a sequence converges to a limit if it converges with respect to every continuous linear functional. This concept is crucial for understanding the behavior of sequences in various mathematical structures, particularly in the context of functional analysis and applications in areas like differential equations and optimization.
Weak vs. weak*: Weak vs. weak* refers to two distinct topologies on dual spaces, where the weak topology is defined on a Banach space and the weak* topology is defined on its dual space. The weak topology relates to convergence based on the behavior of linear functionals, while the weak* topology focuses on convergence through evaluations at points of the original space. Understanding these topologies is crucial for analyzing properties like reflexivity and compactness in functional analysis.
Weak* closed: Weak* closed refers to a property of subsets in the dual space of a normed vector space, where a set is weak* closed if it contains all its limit points under the weak* topology. This concept is crucial for understanding convergence in dual spaces and helps to characterize certain compactness properties, especially in relation to weak* compact sets.
Weak* compact: Weak* compact refers to a set in the dual space of a normed vector space that is compact in the weak* topology, which is defined by the convergence of functionals. In this context, weak* compactness is crucial for understanding the behavior of linear functionals and their limits, particularly in relation to the Banach-Alaoglu theorem, which states that the closed unit ball of the dual space is weak* compact.
Weak* compactness: Weak* compactness refers to a property of sets in the weak* topology on the dual space of a normed vector space. A subset of the dual space is said to be weak* compact if every net or sequence in that subset has a subnet or subsequence that converges to a point in the weak* topology. This concept is closely tied to various aspects of functional analysis, particularly the interplay between bounded linear functionals and topological properties.
Weak* sequentially compactness: Weak* sequentially compactness refers to a property of subsets in the dual space of a normed vector space, where every sequence in the subset has a subsequence that converges in the weak* topology. This concept is crucial for understanding the behavior of bounded sets in dual spaces, as it connects to important results like the Banach-Alaoglu theorem, which states that the closed unit ball in the dual space is weak* compact.
Weak* topology: The weak* topology is a specific type of topology defined on the dual space of a normed space, which allows for the convergence of functionals based on pointwise evaluation rather than norm-based convergence. This topology is crucial for understanding the behavior of linear functionals and their relationships to the original space, particularly when dealing with dual spaces, biduals, and weak convergence.