```Date: Mar 5, 2013 5:44 PM
Author: Kaba
Subject: Re: Orthogonal complement

5.3.2013 1:19, Kaba wrote:>> My intuition says that these properties hold only for finite-dimensional>> spaces, and therefore any proof must necessary use a property which is>> specific to finite dimensional spaces.Ok, I think I got this today. The sequence of Theorems is as follows:1) A vector space V over F is isomorphic to finite(F^I), where I is any set with |I| = dim(V), and finite(.) denotes that subset of F^I which is non-zero only for finitely many i in I.2) The dual space V* of V is isomorphic to F^I, where I is any set with |I| = dim(V).3) If F is a field, then F^I is isomorphic to finite(F^I) if and only if I is a finite set.It follows that4) Vector space V is finite-dimensional if and only if V is isomorphic to its dual space V*.It is this theorem, together with the rank-nullity theorem, which opens up the route to other finite-dimensional theorems, such as the Riesz representation theorem for bilinear spaces, and then to those specific orthogonal complement theorems.I will comment on the other replies later, when I have gone through some more proofs.-- http://kaba.hilvi.org
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