Completeness of orthonormal functions

[Karthiksrao]

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!

 

[fresh_42]

See theorem 1 in
http://math.mit.edu/~jerison/103/handouts/orthonormal.pdf