Lambda-injective normed space ($\Vert \tilde{T}\Vert \le \lambda \Vert T\Vert$)

For $\lambda \ge 1$, a normed vector space $X$ is lambda-injective when every bounded linear operator from a subspace $Y$ of any normed space $Z$ into $X$ extends to $Z$ with the displayed norm bound. The space of bounded scalar functions on an index set is 1-injective: extend each coordinate functional by the Hahn-Banach theorem and reassemble it using the coordinate functional representation of an operator into bounded indexed functions.

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 $J(X)$.