- #1
Karthiksrao
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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!
\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!