I need to provet equality of two definitions of projective module: being direct summand of free module (or equally: having embedding into free module) and having dual basis.
Lets use Wikipedia notation: https://en.wikipedia.org/wiki/Projective_module#Direct_summands_of_free_modules (points 1.3 and 1.5)
I understand that map which takes x to direct sum of $f_{i}(x)$ is an embedding of P into free module which is direct sum or R through indexes $i \in I$, therefore having dual basis implies being direct summand.
I dont understand how second implication from direct summan to dual basis works. https://ncatlab.org/nlab/show/dual+basis On NCatlab these generators ${x_{i}}$ are taken from nowhere (as it often happens on ncatlab).
Okay, as P is direct summand, we have emebedding into free module F = direct sum of R through indexes in I. Okey, let (f_{i}(x)) be embedding of element x.
Of course, for given one specific x we can find such $x_{i}'s$ in P such that $x = sum_{i \in I} f_{i}(x)x_{i}$. But how do i know, that for other element $y \in P$, $y = \sum_{i \in I} f_{i}(y)x_{i}$? I dont get that. Should that dual basis be related with basis of free module F?