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| ``Re: parens''
by rspuzio on 2005-07-01 13:27:34 |
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| > the claim that inner products exist only over the reals and complex
> numbers is disputable.
In order to formulate the notion of an inner product, the only structure one need is the notion of field and the notion of inner product --- put another way, the only terms needed to frame the definition are the primitive terms which occur in the definitions of fields and of vector spaces. Therefore, a restriction that states inner products only exist over the real and complex fields would be artificial. Since nothing in the definition of an inner product requires us to impose restrictions on the field, Ockham's razor says that we should not do so.
In order to formulate the notion of a conjugate-linear inner product, all one requires is an automorphism of order 2. The best known example of such an automorphism is complex conjugation, which is why one usually deals with conjugate-linear inner products over the complex number field, but there is nothing in the definition which inherently limits one to the complex numbers.
Therefore, I would advise that you formulate your definitions in a fashion which does not impose any restriction on the base field except for the requrement that there exist a conjugation of order 2 in the case of the definition of a conjugate linear inner product. Later on, you could mention that the most common case is that of the vector spaces over the fields of real and complex numbers. |
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