The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: countable set of closed subspaces in separable Hilbert space question
Replies: 14   Last Post: Nov 13, 2012 10:29 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 255
Registered: 4/30/07
Re: countable set of closed subspaces in separable Hilbert space question
Posted: Nov 11, 2012 6:51 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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 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


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.