 Solution

Use the convention that the inner product is linear in its first argument. The precise polarization identity isIf (1) holds, apply this formula before and after $U$ to get preservation of all inner products, proving (2). If (2) holds, the identity $⟨Ux,Uy⟩=⟨x,U^{\ast }Uy⟩$ gives $U^{\ast }U=I$. For $z=Ux$, surjectivity then gives $U{U}^{\ast }z=U{U}^{\ast }Ux=Ux=z$, proving (3).

From (3), $\Vert Ux{\Vert }^2=⟨x,U^{\ast }Ux⟩=\Vert x{\Vert }^2$ and likewise $\Vert U^{\ast }x\Vert =\Vert x\Vert$, proving (4). Finally, if (4) holds, apply the same polarization identity to $U^{\ast }$ to obtain $U{U}^{\ast }=I$. This makes $U$ onto; the other norm identity supplies the remaining requirement of (1). ThusThese four statements are conditions in a single equivalence proof, rather than four independent source questions.