Complex inner products

What happens if we want to consider instead of ? The generalization seems to lie right at hand; for and (where now the ’s and ’s may be complex numbers), we write , and we hope that the expressions and can be used as sensible measures of distance. Observe, however, the following strange phenomenon (where ): This means that if is positive, that is, if is at a positive distance from the origin, then is not; in fact the distance from to is imaginary. This is very unpleasant; surely it is reasonable to demand that whatever it is that is going to play the role of in this case, it should have the property that for it never becomes negative. A formal remedy lies close at hand; we could try to write (where the bar denotes complex conjugation). In this definition the expression loses much of its former beauty; it is no longer quite symmetric in and and it is no longer quite linear in each of its variables. But, and this is what prompted us to give our new definition, is surely never negative. It is a priori dubious whether a useful and elegant theory can be built up on the basis of a function that fails to possess so many of the properties that recommended it to our attention in the first place; the apparent inelegance will be justified in what follows by its success. A cheerful portent is this. Consider the space (that is, the set of all complex numbers). It is impossible to draw a picture of any configuration in this space and then to be able to tell it apart from a configuration in , but conceptually it is clearly a different space. The analogue of in this space, for and , is given by , and this expression does have a simple geometric interpretation. If we join and to the origin by straight line segments, will not, to be sure, be the cosine of the angle between the two segments; it turns out that, for , its real part is exactly that cosine.

The complex conjugates that we were forced to introduce here will come back to plague us later; for the present we leave this heuristic introduction and turn to the formal work, after just one more comment on the notation. The similarity of the symbols and , the one used here for inner product and the other used earlier for linear functionals, is not accidental. We shall show later that it is, in fact, only the presence of the complex conjugation in that makes it necessary to use for it a symbol different from . For the present, however, we cannot afford the luxury of confusing the two.