Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Infinite Dimensional Vector Space

  1. Sep 10, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that V is infinite dimensional if and only if there is a sequence v_1, v_2,... of vectors in V such that (v_1,...,v_n) is linearly independent for every positive integer n.

    2. Relevant equations
    A vector space is finite dimensional if some list of vectors in it spans the space.

    3. The attempt at a solution
    We need to prove two directions. For the forward direction, we assume V is infinitely dimensional (and therefore is not finite dimensional). It's really giving me a headache. All of the theorems in my book involve finite-dimensional vector spaces, and none of the proofs seem to give me any information pertaining to the forward direction of this problem. Do I want to use induction somehow? For example, the infinite dimensional vector space F^(infinity) has elements that can be written in the form a_1e_1+a_2e_2+...+a_ne_n+... with the a_i's scalars in the field and the e_i's the vector with all zeroes in every spot except for a 1 in the ith place. That is a linearly independent list, that works for all n, but i don't know how to explain it for every infinite dimensional vector space.

    For the backward direction, first I wanted to assume V is finite dimensional and use contradiction. Then V has a basis that is linearly independent spans V, and has a dimension, lets say m=n-1. Then consider the linearly independent list (v_1,...,v_n). It has dimension n, but every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space, so since dim(V) < dim(v_1,...,v_n) we have a contradiction and V must be infinite dimensional.

    I think the backward direction is almost complete, but i'm having trouble with the forward. Any hints, please?
  2. jcsd
  3. Sep 10, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    V is infinite dimensional. Pick a vector v1 in V. If v1 spanned V, V would be finite dimensional. It's not. So there's another independent vector v2. If {v1,v2} spanned then V would be finite dimensional. It's not. So there's an independent v3. If {v1,v2,v3} spanned then etc. It's sort of induction. The point is that any set of linearly independent vectors can always be extended.
  4. Sep 10, 2008 #3
    Thank you very much, I get it.

    How about the backward direction? Is it correct? Does it need tweaking? Or is it completely wrong?
  5. Sep 11, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    Yeah, I think it's basically ok. The point is that if V has dimension n, then it can't have n+1 linearly independent vectors in it.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?