Results 1 to 2 of 2

Math Help - Subspace of Normed Vector Space

  1. #1
    Member
    Joined
    Mar 2008
    Posts
    148

    Subspace of Normed Vector Space

    Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by iknowone View Post
    Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).
    Let d = \inf\{\|x-c\|:c\in C\} be the distance from x to C. Since C is closed, d>0. For c\in C and 0\ne\lambda\in F, \|c+\lambda x\| = |\lambda|\|\lambda^{-1}c +x\|\geqslant d|\lambda|, so that |\lambda|\leqslant d^{-1}\|c+\lambda x\|. That shows that the map \phi:c+\lambda x\mapsto\lambda is continuous from C+Fx to F.

    Now suppose that (c_n+\lambda_n x) is a convergent sequence in C+Fx, with \textstyle\lim_{n\to\infty}c_n+\lambda_n x = y\in X. (We want to show that y\in C+Fx.) Then c_m-c_n + (\lambda_m-\lambda_n)x\to0 as m,n\to\infty. Deduce from the continuity of \phi that (\lambda_n) is Cauchy in F, hence converges to \lambda_0 say (presumably the scalar field F is meant to be complete). It then follows that c_n\to y-\lambda x = c_0 say, as n\to\infty. But C is closed, so it follows that c_0\in C. Finally, c_n+\lambda_n x \to c_0+\lambda_0x, so y = c_0+\lambda_0x \in C+Fx as required.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subspace of a vector space
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 15th 2011, 10:57 AM
  2. A question on normed vector space
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: July 22nd 2011, 01:19 PM
  3. normed vector space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 14th 2009, 05:21 PM
  4. closure, interior, convex subspace in normed linear space
    Posted in the Differential Geometry Forum
    Replies: 13
    Last Post: November 29th 2009, 06:50 AM
  5. Problem with showing a normed vector space is complete
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 25th 2009, 02:36 PM

Search Tags


/mathhelpforum @mathhelpforum