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I have two questions that are fairly general, but slightly hazy to me still.

1) Can we consider the inner product to be a bilinear function, or not? I would like to think of it as a mapping from an ordered pair of vectors of some vector space V (i.e. VxV) to the field (F), and I know by definition the inner product is conjugate linear as a function of it's first entry (or second, depending on which text you use). But isn't the inner product also linear as a function of either entry whenever the other is held fixed, making it bilinear?

2) Can every mapping from a finite dimensional vector space V to it's field be considered an inner product of something? What about the infinite dimensional case?

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# Inner product space questions

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