Hello,

I have a question, that states that an orthogonal set $\displaystyle \{u_1,u_2,\ldots\}$ is called the backward orthogonalisation of the set $\displaystyle \{x_1,x_2,\ldots\}$ in a Hilbert space, if $\displaystyle \overline{\text{Sp}}\{u_n,u_{n+1},\ldots\} = \overline{\text{Sp}}\{x_n,x_{n+1},\ldots\}$ for all n.

I am requested to find the backward orthogonalisation of the set $\displaystyle \{x_1,\ldots\}$, where $\displaystyle x_n=e_n+e_{k+1}$, where $\displaystyle \{e_n\}$ is the standard basis of $\displaystyle l_2$.

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

Do you have any idea?

Thanks.