Proof of Inner Product in E with Orthonormal Sequence (n=positive integer)

[Poirot1]

let E be an inner product space and (e_n) an orthonormal sequence in E. For x in E and any positive integer n, prove that

Re(<x,(<x,e_1>+...<x,e_k>)e_n>)= |<x,e_1>|^2+...+|<x,e_n>|^2

I got <x,(<x,e_1>+...<x,e_k>)e_n>= <<x,e_1>e_1,x>+...<<x,e_n>e_n,x>

but haven't a clue how to find the real part of this. Sorry for the ugly subscript notation.

 

[Opalg]

Re: inner product proof

let E be an inner product space and (e_n) an orthonormal sequence in E. For x in E and any positive integer n, prove that

Re(<x,(<x,e_1>+...<x,e_k>)e_n>)= |<x,e_1>|^2+...+|<x,e_n>|^2

I got <x,(<x,e_1>+...<x,e_k>)e_n>= <<x,e_1>e_1,x>+...<<x,e_n>e_n,x>

but haven't a clue how to find the real part of this. Sorry for the ugly subscript notation.

This looks wrong to me. Why is there a $k$ on the left side but not on the right? I think that the result should be $$\bigl\langle x,\langle x,x_1\rangle e_1 + \ldots + \langle x,x_n\rangle e_n\bigr\rangle = |\langle x,e_1\rangle|^2 + \ldots + |\langle x,e_n\rangle|^2.$$