On the Scottish Book Problem 155 by Mazur and Sternbach

Problem 155 of the Scottish Book asks whether every bijection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo lspace="0pt">:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> between two Banach spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> with the property that, each point of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> has a neighborhood on which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> is isometric, is globally isometric on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . We prove that this is true under the additional assumption that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is separable and the weaker assumption of surjectivity instead of bijectivity.