Dual space

In mathematics, the dual space is a collection of all possible ways to measure or "test" elements of a given vector space using linear functions, called linear functionals. For any vector, these functionals assign a number in a way that respects addition and scaling. Think of the dual space as a set of tools that analyze or evaluate vectors, much like how sensors or measurement devices assess physical objects. It provides a way to understand the structure of the original space from a different perspective, often revealing important properties and relationships.