What is "inner" about the inner product?

Question

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?

Answer 1

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.