Dual Space

Definition

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.

5 Must Know Facts For Your Next Test

1. The dual space of a normed space is often denoted as $V^*$, where $V$ is the original vector space.
2. The Hahn-Banach theorem provides a way to extend linear functionals from a subspace to the entire space while preserving properties like boundedness.
3. Weak topology involves convergence in terms of dual spaces, where a sequence converges if it converges under all linear functionals in the dual space.
4. The Banach-Alaoglu theorem states that the closed unit ball in the dual space is compact in the weak* topology, which is essential for understanding bounded sets of linear functionals.
5. In reflexive spaces, every bounded sequence has a weakly convergent subsequence, illustrating a deep connection between dual spaces and the geometric structure of Banach spaces.

Review Questions

• How do linear functionals in the dual space relate to the concepts of weak topology and weak convergence?
  • Linear functionals in the dual space are crucial for defining weak topology. A sequence in a normed space converges weakly if it converges under all linear functionals in the dual space. This means that instead of requiring pointwise convergence, we only need convergence when measured by these functionals, providing a broader perspective on convergence in vector spaces.
• Discuss how the Hahn-Banach theorem impacts the structure and properties of dual spaces.
  • The Hahn-Banach theorem allows for the extension of linear functionals defined on subspaces to the entire space without losing boundedness. This theorem significantly influences the structure of dual spaces by ensuring that many desirable properties can be preserved when working with linear functionals. As a result, it provides powerful tools for understanding the relationship between various vector spaces and their duals.
• Evaluate the significance of bidual spaces and their natural embeddings concerning reflexivity in dual spaces.
  • Bidual spaces play a significant role in understanding reflexivity because a Banach space is reflexive if it is naturally isomorphic to its bidual. The natural embedding from a Banach space into its bidual reflects how each vector can be associated with a linear functional acting on it. This connection establishes important properties, like every bounded sequence having a weakly convergent subsequence, thus illustrating deeper relationships between original spaces and their duals.

Related terms

Linear Functional: A linear functional is a linear map from a vector space to its underlying field, typically the real or complex numbers.

Banach Space: A Banach space is a complete normed vector space where every Cauchy sequence converges within the space.

Reflexive Space: A reflexive space is a Banach space that is naturally isomorphic to its bidual space, meaning that the natural embedding of the space into its double dual is surjective.