Results 1 to 5 of 5

Math Help - Infinite dimensional vector Space

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    62

    Infinite dimensional vector Space

    Show that set of real sequence whose infinite sum is convergent is not a finite dimensional vector space over R
    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 math.dj View Post
    Show that set of real sequence whose infinite sum is convergent is not a finite dimensional vector space over R
    For n=1,2,3,..., let S_n be the sequence with a 1 in the n'th position and zeros everywhere else: S_n = (0,0,\ldots,0,\mathop{1}\limits_{\mathstrut\kern-0.5em {(n\text{'th} \atop \text{coord.})}\kern-0.5em},0,0,\ldots).

    Check that these sequences form an infinite linearly independent set (in the vector space of real sequences with convergent sum).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2009
    Posts
    62
    to show that these sequences form a infinite linearly independent set what do i do?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by math.dj View Post
    to show that these sequences form a infinite linearly independent set what do i do?
    For a given natural number N, show that the set \{S_1,S_2,\ldots,S_N\} is linearly independent. That shows that the dimension of the space is at least N. Since that holds for all N, it follows that the space is infinite-dimensional.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Aug 2009
    Posts
    62
    Thank you for your help ...U Rock
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Compact set in metric space (finite and infinite dimensional)
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 30th 2011, 01:38 PM
  2. Stochastic PDE, infinite dimensional Hilbert Space [HARD]
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: May 10th 2011, 08:54 AM
  3. A infinite dimensional linear space
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: April 30th 2010, 08:14 AM
  4. Replies: 1
    Last Post: February 25th 2010, 02:15 AM
  5. Infinite Dimensional Vector Space.
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 15th 2010, 06:59 PM

Search Tags


/mathhelpforum @mathhelpforum