I'm trying to read through the book "Introduction to Global Variational Geometry" by Demeter Krupka and I don't fully understand a proof regarding the existence of a topological complememnt for a subspace with finite codimension. More specifically the Lemma is as following:
I don't understand the part where he says "we choose in the coset $\xi_k$ and arbitrary element $x_k \in X$".
How is $\xi_k$ a coset? Wasn't it a basis element?
Could someone clarify?
Thanks in advance.