Retraction characterization of lambda-injectivity ($PJ=I_X,\Vert P\Vert \le \lambda$)

A space is a lambda-injective normed space exactly when every linear isometry $J:X\to Z$ has a bounded left inverse as displayed. Necessity extends the inverse on $J(X)$. For sufficiency, embed $X$ in bounded scalar functions on an index set, extend the composed operator coordinatewise, and compose that extension with the assumed left inverse. The associated operator $JP$ is a projection onto $J(X)$.