Given $(E,\parallel \; \parallel)$ a Banach space ( complete normed space), and $F\subset E$ a closed linear subspace. There's a natural norm on the quotient linear space $E/F$, and a natural quotient map ( linear and continuous ) :
$P:E\longrightarrow E/F$ with $P(x)=[x]=\{y\in E \; : \; (x-y)\in F \} $ for $x\in E$.
I want to know if it is possible to find a linear continuous map $R:E/F \longrightarrow E$ such that $P\circ R=Id_{E/F}$? Or under what kind of conditions someone could has a positive answer?
