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 (as given in Jackson)

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

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

I am not able to understand 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!

 

[Petr Mugver]

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)

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

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

I am not able to understand 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!

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:

$f=\sum_nU_n(U_n,f)$

or, in the x-representation

$f(x)=\sum_n U_n(x)\int U^*_N(x')f(x')dx'$

but we also have

$f(x)=\int f(x')\delta(x-x')dx'$

so...