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Infinite dim vector space

  1. Dec 14, 2005 #1
    We have x=x1(1,0,0) + x2(0,1,0) + x3(0,0,1) to represent R^3. That's a finite dimensional vector space. So what do we need infinite dimensional vector space for? Why do we need (1,0,0,...), (0,1,0,0,...), etc. bases vectors to represent R^1 ?
  2. jcsd
  3. Dec 14, 2005 #2


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    To represent R^1, you only need 1 vector. A vector space of dimension n is spanned by a basis of n vectors, just as in your example of R^3. This is because a basis needs to span the vector space (which means you need *at least* n vectors) and has to be linearly independant (which means you can only have *at most* n vectors) which makes the number of vectors in the basis exactly n.

    Then, what do we need vector spaces with infinite dimension? Consider the vectorspace [itex]\mathbb{R}\left[ X \right][/itex] which is the vector space of all polynomials in x over R. This is trivially an infinite dimensional vector space since a finite number of vectors in a basis contains a vector with a maximum degree r, meaning that x^(r+1) and higher cannot be formed.
  4. Dec 14, 2005 #3


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    "Functional Analysis" makes intensive use of "function spaces"- infinite dimensional vector spaces of functions satisfying certain conditions. TD gave a simple example- the space of all polynomials. Perhaps the most important is L2(X), the vector space of all functions whose squares are Lebesque integrable on set X.
  5. Dec 14, 2005 #4


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    Another familiar example of an infinite dimensional vector space is functions from an infinite domain to a ring
    Consider that the space of functions
    [tex]f:A \rightarrow R[/tex]
    from some set [itex]A[/itex] to a ring [itex]R[/itex]
    is a vector space with dimensions indexed on [itex]A[/itex]
    since we have a vector
    Scalar multiplication, and vector addition are performed using the ring.
    Last edited: Dec 14, 2005
  6. Dec 14, 2005 #5

    matt grime

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    I think you should reconsider that example, NateTG. How is *a* function a vector space? Over what field? And what are its elements
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