Hello,
I have a question, that states that an orthogonal set \{u_1,u_2,\ldots\} is called the backward orthogonalisation of the set \{x_1,x_2,\ldots\} in a Hilbert space, if \overline{\text{Sp}}\{u_n,u_{n+1},\ldots\} = \overline{\text{Sp}}\{x_n,x_{n+1},\ldots\} for all n.
I am requested to find the backward orthogonalisation of the set \{x_1,\ldots\}, where x_n=e_n+e_{k+1}, where \{e_n\} is the standard basis of l_2.

I am pretty sure I should be able to express u_k in the form x_k+\sum_{j=k+1}^\infty a_jx_j for all k, but that quickly leads me to a_n=\pm 1 for all n, and that can't be it.
Do you have any idea?

Thanks.