Results 1 to 2 of 2

Thread: Infinite Dimensional Vector Spaces

  1. #1
    Senior Member Sampras's Avatar
    Joined
    May 2009
    Posts
    301

    Infinite Dimensional Vector Spaces

    Suppose you have a infinite dimensional vector space $\displaystyle V $. Suppose $\displaystyle T \in \mathcal{L}(V,W) $. Can we have $\displaystyle \dim V = \dim \text{null} \ T + \dim \text{range} \ T$?

    E.g. can we partition an infinite dimensional vector space into finite dimensional ones?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    I'm not sure how your two questions are related. The dimension theorem holds for infinite-dimensional vector spaces; but all that this tells us is that if $\displaystyle \dim V = \infty$ then either the range or the kernel of $\displaystyle T$ must be infinite-dimensional.

    It is impossible to partition a vector space into subspaces. If $\displaystyle W_1, W_2$ are subspaces and $\displaystyle W_1 \cup W_2 = V$ then we must have $\displaystyle W_1 \subset V$ or $\displaystyle W_2 \subset V$. On top of that, the zero vector would have to be in every part.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: Jan 3rd 2012, 12:56 PM
  2. Infinite Dimensional Banach Spaces
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Mar 23rd 2011, 11:46 AM
  3. [SOLVED] Infinite Dimensional Vector Spaces and Linear Operators
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 9th 2011, 02:28 AM
  4. Infinite dimensional spaces and other questions.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 26th 2011, 06:05 AM
  5. Infinite-dimensional vector spaces and their bases!
    Posted in the Advanced Algebra Forum
    Replies: 12
    Last Post: Jul 23rd 2010, 01:42 PM

Search Tags


/mathhelpforum @mathhelpforum