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I'm trying to fix in my head a very precise definition of what to mean for an euclidean space, as we use it in multivariable calculus.

The def. I had in my mind was that an ES is

(1) a real vector space

(2) of finite dimension

(3) with the "standard" (dot)

(4) inner product

I'm pretty sure of (1) and (4) (it has to be some vector space, and has to have an inner product to mesasure distances and angles). Not so sure about (2) (maybe infinite dimensional euclidean spaces are well defined?) and (3) (I've read somewhere it is not necesary that the inner product of an ES be the "dot" one, but changing the inner product doesn't change all measures (distances and angles) in the space? Does all theorems about ES still apply if I change the inner product?)

But what more bothers me is that there is no definition of a "cross product" in that definition, so I should not use it when calculating "surface integrals" in euclidean space [itex] \mathbb{R}^3 [/itex].

Or is there a way of calculating [itex] a \times b [/itex] using only the inner product [itex] a \cdot b [/itex] ?

A little confused with those things.

Any help?

Thanks,

Damián.

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# Euclidean Space definition

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