I´m looking for an online document where I can find information explaining the construction of vector and tensor representations of SO(N). Does anyone can indicate any?
The representations of SO(n) will be subsumed under the more general area of Lie Groups, since they will be weight representations you're probably interested in. So don't be too specific in your search.
It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC).
Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?