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Does anyone know or can anyone give a reference explaining how the concepts of a linear functional and particularly that of a dual space developed? I know Riesz published his famous representation theorem in the first decade of the $1900$s, but did he use these concepts then?

My main reason for asking this question is motivation: These two concepts seem to arise out of thin air in all linear algebra books that I have looked at. It would be nice to find a motivating (non-contrived) example that forces you to look at the dual space of a vector space.

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    Linear functionals don't really "come out of nowhere". Just as we can consider normal vectors to be column matrices (n x 1), we can do the same for "row matrices" (1 x n). Just as we can consider matrices as representing a linear transformation, the inner product arises naturally (in this same view) as $x^Ty$. What happens when we view the standard basis column vectors as row vectors? Then $e_j$ becomes the linear functional $\pi_j$, which projects a vector onto its j-th coordinate.2012-07-02
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    If I were exploring the difference between row vectors and column vectors, I would make two observations: row vector x column vector = scalar and column vector x row vector = matrix. That this would lead me to the concepts of linear functional or dual space seems unlikely to me.2012-07-02
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    Dear echoone, [This answer](http://math.stackexchange.com/a/3755/221), while at an elementary level, may still shed some light. Regards,2012-07-03
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    The correspondence between linear transformations from an n-dimensional vector space to an m-dimensional vector space (over a field F) and the vector space of all mxn matrices with entries in F is often explictly or implicitly covered in most linear algebra courses. Much time is typically devoted to the linear transformation induced by an mxn matrix $A$, where $A$ acts on an nx1 column vector $x$ to produce an mx1 column vector $Ax$. Linear functionals (in this view) are just the special case $m = 1$.2012-07-03
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    @MattE I like that explanation. In fact, that is exactly how it was done historically according to the stuff I have read.2012-07-04
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    Did someone, say Gauss, know of this duality? Specifically the duality between a vector space and linear transformations over it.2013-10-23

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