A dual space is a mathematical concept in linear algebra that consists of all linear functionals defined on a vector space. If you have a vector space V, its dual space, denoted as V*, includes all functions that take a vector from V and return a scalar, typically from the field of real or complex numbers. These functionals are linear, meaning they satisfy properties like additivity and homogeneity. The dual space plays a crucial role in various areas of mathematics and physics, particularly in functional analysis and quantum mechanics. It helps in understanding the relationships between vectors and their corresponding linear functionals, providing insights into the structure of V and its transformations.