Who was the first person who used the dual space? In which paper / book did he or she use the dual space?
Who was the first who called it dual space and in which paper / book?
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It was Hahn (of Hahn-Banach Theorem) who, in 1927, first introduced the dual of a normed linear space ("polare Raum" was his term.) Hahn had extended previous arguments for separable spaces in order to obtain the existence of general continuous linear functionals on a normed linear space; Hahn did this by introducing methods of transfinite induction. Hahn introduced the idea of regular normed spaces (now called reflexive.) It makes sense that the existence of general classes of continuous linear functionals would lead to a proper notion of a dual space.
Two years later after Hahn, Banach proved the same theorem with the same proof (He later acknowledged the primacy of Hahn's work.) Banach introduced a convex functional and extensions of linear functional bounded by convex functions, which paved the way for locally-convex spaces.
It appears that the understanding gained in Functional Analysis came before a proper understanding of the dual for a finite dimensional vector space! In History of Functional Analysis, J. Dieudonne writes that "before 1930 nobody had a correct conception of duality between finite dimensional vector spaces; even in van der Waerden's book (1931), such a vector space and its dual are still identified. All this was to weigh heavily on the evolution of a linear Functional Analysis; in particular it followed (over a shorter span of years) the same unfortunate succession of stages through which linear Algebra had to go; and it is only after it was realized that the current conception of vectors as "n-tuples" could not possibly be extended to infinite dimensional function spaces, that this conception was finally abandoned and that geometrical notions won the day."
Stefan Banach in the "Théorie des opérations linéaires" is a good bet.