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Math Help - Inner product space without less than relation

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    Inner product space without less than relation

    Hey, so the cauchy swartz inequality seems dependent upon having a less than relation defined on your innerproduct space, but I have never seen that such a relation is nessascary in the definition of an inner product space. So my question is, can you come up with an example of an inner product space where the cauchy swartz inequaility does not apply?
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    Quote Originally Posted by Chris11 View Post
    Hey, so the cauchy swartz inequality seems dependent upon having a less than relation defined on your innerproduct space, but I have never seen that such a relation is nessascary in the definition of an inner product space.
    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 presupposed ordered field.

    So my question is, can you come up with an example of an inner product space where the cauchy swartz inequaility does not apply?
    You would have to come up with an inner product space over a field of scalars that is not ordered. The definitions of "inner product space" that I have seen always immediately limit the field of scalars to \mathbb{R} or \mathbb{C}, 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 \geq 0 , so there you go: requiring that ordering relation between scalard already in the very definition of an inner product.
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    Quote Originally Posted by Chris11 View Post
    Hey, so the cauchy swartz inequality seems dependent upon having a less than relation defined on your innerproduct space, but I have never seen that such a relation is nessascary in the definition of an inner product space. So my question is, can you come up with an example of an inner product space where the cauchy swartz inequaility does not apply?
    This may not be quite what you were getting at, but there are spaces having a structure that is sort of like an inner product but which is not positive definite. An important example is the Minkowski "inner product" in the four-dimensional space-time continuum, in which the "distance" between two points is zero if they lie on the same light cone.
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