The Dual Space

The dual space of a vector space is the collection of all linear functionals—these are functions that take vectors and assign each a single number in a way that respects addition and scalar multiplication. Think of it as a set of measurement tools tailored to the original space, where each tool provides a numerical "assessment" of any vector’s properties. The dual space itself forms a vector space, reflecting the structure of the original space but focusing on how vectors can be evaluated rather than combined. It’s fundamental in areas like functional analysis and quantum mechanics, connecting vectors to meaningful numerical interpretations.