# Why can a function satisfying same boundary conditions as functions Un(x) of a complete set be expanded as Î£CnUn(x)?

This is a question I encountered while reading

"It can be shown that a function

...

If we multiply by U*m(x) and integrate, we obtain

<

This second part is fairly straightforward because

∫

and it follows that

<

However, the first part confuses me because I can only justify it when

In that case, it can easily be seen that the expression

Thank you in advance for answering! Please feel free to post any questions of your own if you need clarification. Also, please be advised that you should not post responses on the order of "you didn't specify an interval of integration, so your question is impossible!" - if you know enough to answer the question, then you will understand why said interval is not specified, etc. Once again, thanks!

*Introduction to Quantum Theory*by Hendrik F. Hameka (Please don't worry, I'm not trying to cheat on my homework, I am just asking out of curiosity). The book states the following:"It can be shown that a function

*f(x)*that satisfies the same boundary conditions as the functions*U*n*(x)*of a complete set can be expanded as*f(x)*=**Σ***C*n*U*n*(x)*...

If we multiply by U*m(x) and integrate, we obtain

<

*U*m |*f*> =*C*m "This second part is fairly straightforward because

∫

*U**m*f(x)*dx = ∫*U**m**Σ***C*n*U*n*(x)*dxand it follows that

<

*U*m |*f*> =*C*n**Σ**<*U*m |*U*n > =*C*n**Σ**δn,m =*C*nHowever, the first part confuses me because I can only justify it when

*U*n*(x)*= (2π)^(-1/2) e^(inx) (The example of a complete set given in the book, actually)In that case, it can easily be seen that the expression

**Σ**<*U*n |*f*>*U*n*(x)*is just the Fourier Series expansion of*f(x)*and thus the result is valid for the particular complete set. However this does not explain why this is valid for all complete sets:**"how can you prove that this conclusion is valid not only for this particular complete set, but for all complete sets?"**is my question.Thank you in advance for answering! Please feel free to post any questions of your own if you need clarification. Also, please be advised that you should not post responses on the order of "you didn't specify an interval of integration, so your question is impossible!" - if you know enough to answer the question, then you will understand why said interval is not specified, etc. Once again, thanks!

Mathematics for Quantum Mechanicsby John David Jackson and it went into some more depth by (I think) using Lagrange Multipliers in order to minimize the mean square error of the original sigma expression in order to derive Bessel's Inequality and then show that the coefficientCn must equal <Um |f> when any series of functions from a complete set is used. However, I am still a bit confused and the math is starting to make my head spin. Does anyone think they can help to explain exactly what is going on, step by step and with full mathematical rigor?