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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

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

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

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

    Why should the summation for the functions at two different points go to zero ? 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
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