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The inner product I am asking about is the one that generalizes the dot product for an arbitrary inner product space. Why is it called an "inner" product? Is there an outer product? Who named it so, and when?

MJD
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As far as I remember, Grassmann introduced different products in his ("unreadable work" as Moebius stated) "Die lineale Ausdehnungslehre". In particular, he defined "die Aeussere Multiplikation der Strecken" (outer multiplication of line segment) and also die internal, a.k.a. inner version of it. This is discussed, for example, in "Grassmann" by H.S.Petsche. I think that originally Grassmann worked in a pure geometrical setting, with lines, curves and surfaces, and only later the products have been translated in the linear algebra context.

EDIT: Origin of the terminology of inner product. Grassman studied Leibniz's theory of congruence of line segments.He translated the study of congruence of line segments $ab$ into the study of certain functions $f$ of the vectors $a-b$ defined by the line segments themselves. He arrived at the conclusion that these fucntions must satisfy $f(a-b)=f(b-a)$, and that $f(a-b)$ is equal to the length of $ab$; in other words he defined an inner product of $a-b$ with itself. The terminology "inner products" is firstly referred to the "Inneren Produkten je zweier paralleler Strecken" (inner product of any 2 parallel line segments) and then extended to non-parallel ones.

Avitus
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    After reading this I still don't know what's inner about the inner product. Is it because it originates as a multiplication of something with itself? Or because it's related to a line segment between two points (the inner part of the line)? Or something else? – isarandi Mar 20 '15 at 12:24
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    @isarandi The inner product as per Grassmann himself is called inner because it grows as two vectors become more parallel and the outer grows as they become more distant, or more perpendicular. See here for a translation from Grassmann: https://twitter.com/michaelbachtold/status/1111975591320961025?s=19 – Cristian Em. Mar 30 '19 at 16:16
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    I always remembered it by the position of the T (transpose) symbol. For the usual case of column vectors, the inner product has the T in between the two operands (as the first operand is transposed), and for the outer product the T is on the second operand so it's "on the outside". But it seems like this is just a superficial mnemonic device and has no relation to the origins of the terminology. – isarandi Mar 30 '19 at 17:41
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    How does “inner” get associated with parallel? I read the link you included @CristianEm. and I personally don’t understand any better. – gen-ℤ ready to perish May 04 '20 at 00:01
  • @gen-zreadytoperish Okay, I reread the image. If we strictly mean "outer" to be "kronecker product", then that image is not relevant. See the comment "In a nutshell: uHv - inner, uvH - outer ;-)" from AlexR which makes sense. If by "outer" we mean exterior/wedge product then it makes more sense. I think of "inner" and "exterior" to refer to when the product is biggest in magnitude. For "exterior", it's basically the determinant, as this is the most nonzero when the vectors are perpendicular. The inner product is most nonzero when the vectors are parallel/proportional. I hope this helps. – Cristian Em. May 10 '20 at 13:55
  • @gen-zreadytoperish It's just a name in the end. I mentally associate to "inner" words such as "closeness" or "alikeness", so it makes sense to me for the inner product to measure how alike two vectors are in direction. For me "outer" or "exterior" gives me the feeling of measuring how spread apart something is, and the exterior product measures how linearly independent two or more vectors are. Note what I said is in no way relevant to how historically this arrived, but I realized that if other people have different connotations for the words then these labels might make less sense for them. – Cristian Em. May 10 '20 at 14:00