On Sun, 11 Nov 2012 03:51:24 -0800 (PST), "Daniel J. Greenhoe" <firstname.lastname@example.org> wrote:
>On Sunday, November 11, 2012 5:09:34 PM UTC+8, William Elliot wrote: >> ...Do you mean the topological closure or some algebra construction? > >There is no such thing as "algebraic closure" in mathematics.
There most certainly is.
Which is not to say your meaning was unclear - William likes to misunderstand things for no good reason.
Otoh you do need to explain the difference between limit and union here...
>There is only topological closure. >Closure is always with respect to a topology. >A norm can induce a topology, and norms do have a powerful algebraic structure, but it is the topology induced by the algebraic structure of the norm that defines closure. > >> How are you defining lim(n->oo) X_n? > >"Strong convergence" ("convergence in the norm"); that is, the norm induced by the inner product: > >For any e>0 there exists N such that > || x-x_n || < e for all n>N > >Dan