(What i'm about to say is a subpart of a proof of a theorem).
Let $M, N$ be a closed subspace and a finite dimensional subspace, respectively, of a normed linear space $X$. We define the natural map $A$ from $X$ onto $X/M$ by $Ax = [x]$.
Here the question... i think it is safe to state that $AN = \left\{ n + M \;|\; n \in N\right\}$ but i don't get why $A^{-1} A N = M + N$ ($A^{-1}$ is used in the set theoretic sense).
I mean since $[n] = n + M$ then i would be tempted to say that $AN = N + M$ how could it be the "inverse" be the same?
I mean i understand the meaning by intuition but i don't get the "formal meaning".
Could you help me to have a more thorough look into this?
I think you have a problem of notation. I mean $AN$ is in effect a $X/M$ subspace while $ { n+M| n\in N}=N+M$ is a subspace of $X$. So $AN={ n+M| n\in N}$ it's formally uncorrect and you should write $A^{-1}AN={ n+M| n\in N}$.