Completeness of orthonormal functions

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The discussion centers on the completeness of orthonormal functions in function space, particularly in the context of expansions used in fields like electrodynamics. The completeness condition is expressed mathematically, indicating that the sum of products of orthonormal functions at two different points equals the Dirac delta function. This raises questions about the intuition behind why the summation approaches zero for non-equal points and the implications for spanning the entire space. Additionally, there is a request for a parallel explanation in vector space to enhance understanding of the concept. The inquiry highlights the need for clarity on the relationship between orthonormal functions and their completeness in representing functions across a defined space.
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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

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

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!
 
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