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

Span of an infinite set

  1. Sep 30, 2013 #1
    Suppose that some infinite set S spans V. Then this means every vector in V is expressible as some linear combination of the vectors in S. Does this combination have to be finite?

    It couldn't be infinite, because that necessarily invokes notions of convergence and norm which do not necessarily apply to an arbitrary vector space?

  2. jcsd
  3. Sep 30, 2013 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    That's correct. Assuming you have a well defined notion for them, the set of infinite linear combinations is what's called the completion of the span of S.
  4. Oct 1, 2013 #3


    User Avatar
    Science Advisor

    You haven't defined V well enough. If V has a topology, then completeness is meaningful.

    Without a topology you need to be very precise in defining "spans". It may mean that every vector in V is expressible by a finite linear combination. Essentially the question is answered by "yes" by definition.
  5. Oct 1, 2013 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    A bit of terminology may be in order.

    For an infinite-dimensional vector space, a Hamel basis refers to a basis in the linear algebra sense: every element of the vector space can be expressed uniquely as a linear combination of a FINITE number of elements of the Hamel basis. Every vector space has a Hamel basis, as a consequence of Zorn's lemma, but in general it's not possible to specify one concretely.

    As others have noted, if you want to allow infinite linear combinations, there needs to be a topology involved. For an infinite-dimensional topological vector space, one has the notion of a Schauder basis: http://en.wikipedia.org/wiki/Schauder_basis But not every topological vector space necessarily has such a basis. If you impose more structure, then you can have a guarantee: for example, every Hilbert space has a basis in this sense (orthonormal, even).
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook