The ordering relation is not usually made an explicit part of the definition of the inner product space, because it is defined for the scalars of the presupposedorderedfield.

You would have to come up with an inner product space over a field of scalars that isSo my question is, can you come up with an example of an inner product space where the cauchy swartz inequaility does not apply?notordered. The definitions of "inner product space" that I have seen always immediately limit the field of scalars to or , and that eliminates any room for the kind of example you are asking for.

Also, you want an inner product to have "positive definiteness", which means that the inner product of a vector with itself is always , so there you go: requiring that ordering relation between scalard already in the verydefinitionof an inner product.