relations between distance-preserving, norm-preserving, and inner product-preserving maps

Solution 1:

I am not completely sure what your question is. Here are some thoughts which seem to be relevant:

Let E be a normed space with induced metric, and $f:E\to E$ a set-map.

• If f preserves distance and f(0)=0, then it preserves norm: $\|fv\|=d(fv,f0)=d(v,0)=\|v\|$.
• If f preserves norm and addition, then it preserves distance: $d(fv,fw)=\|f(v-w)\|=\|v-w\|=d(v,w)$.
• If f preserves distance and is surjective, then it is affine, by Mazur–Ulam (thus linear if moreover $f(0)=0$).

Let F be an inner-product space, with induced norm and metric, and $f:F\to F$ a set-map.

• If f is linear, then to preserve distance, norm, or inner-product are three equivalent conditions: for distance and norm this follows from the above, and for norm and inner product this follows by polarization.