# Who was the first to use dual space?

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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See this question for some comments. – Per Manne Aug 29 '12 at 14:24
Do you mean the "algebraic dual" of all linear functionals or the "topological dual" of just the continuous ones? – rschwieb Aug 29 '12 at 14:42
@rschwieb: I mean the "algebraic dual". – Martin Thoma Aug 29 '12 at 15:29
@moose Hmm, that might go back at least as far as the development of vector analysis... I have no idea! – rschwieb Aug 29 '12 at 16:12
So ... 19th century physics has "vectors" and "co-vectors". – GEdgar Sep 28 '12 at 20:03

## 2 Answers

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."

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Stefan Banach in the "Théorie des opérations linéaires" is a good bet.

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I don't think this is a good bet (even if restricting the question to the functional analytic dual). The Riesz representation theorems (functionals on Hilbert spaces) were published in 1907 by both Fréchet and Riesz, while the Riesz representation theorem on measures on the unit appeared in 1909. – t.b. Aug 30 '12 at 8:25