Results 1 to 3 of 3

Math Help - Dual Space of a Vector Space Question

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Dual Space of a Vector Space Question

    Hello everyone I have a question about Dual Spaces. Let X be an infinite dimensional vector space over \mathbb{C}, and let
    \{ x_1, \ldots, x_n \} be a subset of X, which is linearly independent. Then we can define the unique linear functionals $f_i : X \rightarrow \mathbb{C} which give  f_i (x_j) = 1 if  i=j, 0 otherwise.

    My question is can we choose the f_i's to be continuous? I was asked this question and I am not sure what the answer is. I take it if we have an infinite basis \{ x_i\}_{i \in I} of X then we cannot have every f_i continuous because then surely every linear functional would then be determined by those continuous functions, i.e. it is a linear combination of finitely many of them, hence is continuous. And there exist unbounded linear functionals (?). Any help with this would be appreciated! Thanks
    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 slevvio View Post
    Hello everyone I have a question about Dual Spaces. Let X be an infinite dimensional vector space over \mathbb{C}, and let
    \{ x_1, \ldots, x_n \} be a subset of X, which is linearly independent. Then we can define the unique linear functionals $f_i : X \rightarrow \mathbb{C} which give  f_i (x_j) = 1 if  i=j, 0 otherwise.

    My question is can we choose the f_i's to be continuous? I was asked this question and I am not sure what the answer is. I take it if we have an infinite basis \{ x_i\}_{i \in I} of X then we cannot have every f_i continuous because then surely every linear functional would then be determined by those continuous functions, i.e. it is a linear combination of finitely many of them, hence is continuous. And there exist unbounded linear functionals (?). Any help with this would be appreciated! Thanks
    The conditions  f_i (x_j) = 1 if  i=j, 0 otherwise, define f_i uniquely as a continuous linear functional on the finite-dimensional subspace spanned by x_1, \ldots, x_n. The functional can then be extended to a linear functional on the whole of X. The extension will not be unique. The functional can in fact be extended to a continuous linear functional on the whole of X, by the Hahn–Banach theorem. This continuous extension will also not be unique.

    If the set \{ x_i\} is infinite, then the functionals  f_i need not be continuous, even on the subspace spanned by the \{ x_i\}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Thank you! I looked at the Hahn Banach theorem and understand what's going on a little better
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question on null space/column space/row space of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 1st 2011, 01:47 PM
  2. Normed space and dual space
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: June 5th 2011, 10:46 PM
  3. Dual space of a vector space.
    Posted in the Advanced Algebra Forum
    Replies: 15
    Last Post: March 6th 2011, 02:20 PM
  4. closed in dual space E* implies closed in product space F^E
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 21st 2010, 04:58 AM
  5. vector space and its dual space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 26th 2009, 08:34 AM

Search Tags


/mathhelpforum @mathhelpforum