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Completeness of orthonormal functions

  1. Jul 26, 2011 #1
    In many areas (say, electrodynamics) we come across expansions of any function in terms of a series of orthonormal functions that span the space. Now the condition for completeness of a set of orthonormal functions in that space is given by (as given in Jackson)

    [tex]\sum_{n=1}^\infty U_n^*(x') U_n(x) = \delta(x'-x)[/tex]

    where x and x' are two points in the function space.

    I am not able to understand what this is intuitively due to.

    Why would the orthogonal functions not span the entire space if the summation does not go to zero ?

    Finally, is there a corresponding relation in vector space ? That will probably give me a better understanding of what is happening, if we extend it to function space..

    Thanks a ton!
  2. jcsd
  3. Jul 27, 2011 #2
    I strongly suggest you to study the Dirac's formalism of bras and kets, it makes all theese relations much clearer.

    Anyway, if U_n is complete and orthonormal, you must have, for every f belonging to the space:


    or, in the x-representation

    [itex]f(x)=\sum_n U_n(x)\int U^*_N(x')f(x')dx'[/itex]

    but we also have

    [itex]f(x)=\int f(x')\delta(x-x')dx'[/itex]

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