I'm self-learning functional analysis at the moment and although I can understand the underlying theory I have difficulty applying it in the aggregate.
Can someone please break this down for me step by step, It would also be very helpful if you could also add some kind of thought process. This is an example of the type of question I'm looking at:
$$f \begin{pmatrix} \xi_1 \\ \xi_2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1\end{pmatrix} + \begin{pmatrix} \frac25& \frac45 \\ 0 & \frac25 \end{pmatrix} \begin{pmatrix} \xi_1 \\ \xi_2 \end{pmatrix}$$
Show that this is a contraction mapping on $\ell^2(2)$ but not $\ell^1(2)$
What is the fixed point of this mapping?
Thanks!