Well, OK, maybe my notation doesn't bother you as much as it does me.

Here's the thing: elements of the Cartesian products of sets, say AÃ—B are the ordered pairs (a,b), a from A, b from B. In an inner product space, the scalar (v,w) is certainly an ordered pair, an element of VÃ—V. But not all spaces have an inner product defined, so if, for one such space, I write (x,y) as an element of WÃ—W we can take it that (x,y) is not an inner product.

I'm going to change notation for a bit, and write [v,w] when I want to specify that this is an inner product. Where it makes no difference, I will simply use (v,w), which may or may not be an inner product, it's irrelevant for the purpose at hand (this, I think, will be usually).

The other problem that may have escaped you is this. I'm writing f_{v}for the element of V*, the vector dual to the vector v in V. Suppose, though, I now say that f_{v}(v) = ||v|| for all v in V, it isn't clear that, in this case, I must insist that f[/sub]v_{1}[/sub] is dualonlyto v_{1}, which is a very cumbersome way to proceed. But here I don't see an easy way out.

We'll have to live with it, so let me emphasize: the "f" has no real meaning, and simply means "some functional" that takes V → R. As the spaces V and V* are isomorphic (provided only that V is finite), this implies that foreachv in V there is oneand only onef_{v}in V*. Maybe you already made that leap, if so, good on ya.

And finally: the form f_{v}⊗f_{w}: VÃ—W → R I wrote earlier was for illustration only, and not a good illustration at that - I urge you to erase it from your mind, it muddies the waters.

Tensors are formed exclusively from the multilinear elements of the space V⊗...⊗V⊗V*⊗...⊗V* acting on V*Ã—...Ã—V*Ã—VÃ—...Ã—V, i.e multiple copies of some space and its dual.