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An inner product can induce a norm by defining $\|x\| = \sqrt{\langle x,x \rangle}$, the a norm can induce a metric by setting $d(x,y) = \|x - y\|$. But not every norm (metric) is induced from inner product (norm), unless the parallelogram law (homogeneity and translation invariance conditions) is (are) satisfied.

Suppose an inner product induces a norm, which then induces a metric, using these defined inner product, norm and metric, can we tell what are the relations between distance-preserving, norm-preserving, and inner product-preserving maps? I just know one: if an isometry (distance-preserving), which is injective, is also surjective, then it's unitary (bijective), which means the isometry is also inner product-preserving. For example, distance-preserving maps on a compact metric space are also inner product-preserving. Are there any other relations?

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In the last sentence, how do you define inner product on compact metric space? –  Norbert Jul 12 '12 at 17:40
    
@Norbert I guess it becomes dot product. Correct me if I'm wrong. –  chaohuang Jul 12 '12 at 17:51
    
What is dot product? As for inner products they are defined only for linear spaces. –  Norbert Jul 12 '12 at 18:10
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up vote 5 down vote accepted

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.
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"If f preserves norm and addition,..." correctly: "If f preserves norm and subtraction,..." –  user63284 Feb 21 '13 at 8:48
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