Tingley's problem: does every surjective isometry between unit spheres of Banach spaces extend to a linear isometry?

Theorem 1.1: if $X,Y$ are nonzero real Banach spaces and $f:S_X\to S_Y$ is a surjective isometry, then $T(0)=0$, $T(x)=\|x\|f(x/\|x\|)$ is a surjective real-linear isometry, the unique linear extension of $f$. No separability, reflexivity, smoothness, strict convexity or finite-dimensionality is assumed. For complex spaces only real linearity is concluded. The proof uses a maximal-defect argument, common supports and Darbo's fixed-point theorem. Incomplete normed spaces are not treated.

Theorem 1.1: if $X,Y$ are nonzero real Banach spaces and $f:S_X\to S_Y$ is a surjective isometry, then $T(0)=0$, $T(x)=\|x\|f(x/\|x\|)$ is a surjective real-linear isometry, the unique linear extension of $f$. No separability, reflexivity, smoothness, strict convexity or finite-dimensionality is assumed. For complex spaces only real linearity is concluded. The proof uses a maximal-defect argument, common supports and Darbo's fixed-point theorem. Incomplete normed spaces are not treated.