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