Dual space

The dual space (or continuous dual) of a normed vector space X is the space X* of all continuous linear functionals on X — the linear maps from X to the scalar field (ℝ or ℂ) that are bounded with respect to the norm. The dual space is itself a Banach space under the operator norm, and this completeness is automatic regardless of whether X is complete. The dual is the lens through which the geometry of X is studied: the Hahn-Banach theorem guarantees that X* is rich enough to separate points, and the embedding of X into X** is the gateway to the theory of reflexivity. The dual space is not a derivative construction; it is the primary space in which the structure of X is encoded.