No, stephencanon's comment is what I mean, and this may be an example of what my prof. meant that my students would not then be able to teach calculus. By that

No, stephencanon's comment is what I mean, and this may be an example of what my prof. meant that my students would not then be able to teach calculus. By that point (this course was "differential operators", 700-level stuff) we all had a decent intuition of what inner and outer products are. The prof's comment was that looking at f(x) as an infinite-dimensioned vector, there is a unit *-cube g(x)=1 of compatible dimensions that can produce an inner product f|g (I'm not going to hunt through my character map for the dot or the integral sign). That inner product is the same as Integral(f(x), dx). This was in analogy to the differential operator being the exterior ("wedge") product of a function and its field. The real point of the definition was relating y' and Integral(y) to div y and grad y: in y' you're going from vectors to tensors, and in Integral(y) you're going from vectors to scalars. Or, Integral(y) is a projection of y on some unit cube, and y' is finding the function of which y is the projection on an appropriate unit cube.