I look for an advanced linear algebra (A complete book but wich deals indiferently with infinite/finite vector space). To give an idea i expect a book that (for exemple) would prove the existence of a base in any vector space by Zorn Lemma.


PS: I found Bourbaki's book but the level is quite hard

  • $\begingroup$ Maybe you whish to add the tag "soft question". Apart from that you are looking for two different things since linear algebra studies finite dimensional vector spaces. The introduction subject to infinite dimensional vector spaces is functional analysis. $\endgroup$ – noctusraid Feb 8 '16 at 15:00
  • $\begingroup$ I don't necessarily think you should judge the quality of a book on linear algebra by whether it includes that particular proof. For purely algebraic treatments of linear algebra, that may be the only place that Zorn's lemma is needed, and authors may quote the result without proof to avoid what they consider a digression. With that in mind, I think Vols. 1 and 2 of Cours de mathématiques spéciales by Ramis and Cours d'algèbre by Godement are good places to learn linear algebra. But Lang's Algebra and Jacobson's Basic Algebra II (Ch. 3 on modules) both contain this proof and much more. $\endgroup$ – David Feb 8 '16 at 21:26
  • $\begingroup$ Thanks a lot David. Actually I work with RDO and i was looking for small complements to distinguish (when it is important) infinite and finite dimensional spaces. $\endgroup$ – curious Feb 11 '16 at 9:38

The second volume of Jacobson's Lectures in abstract algebra (in particular, Chapter VIII on infinite-dimensional vector spaces) could be a good reference.

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