Relations between finite dimensional normed vector space and inner product space

Asked Dec 09 '23 at 12:20

Active Dec 09 '23 at 14:41

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This is from A Course in Metric Geometry by D. Burago, Y. Burago, and S. Ivanov, exercise 1.2.24 p.7.

Let $V$ be a finite-dimensional normed vector space. Prove that $V$ is Euclidean if and only if for any two vectors $v,w\in V$ such that $|v|=|w|$, there exists a linear isometry $f:V\rightarrow V$ such that $|w|=|f(v)|$.

My question is that I have no idea to prove the "if". When $V$ is an inner product space, then the existence of $f$ is just linear algebra. I want to obtain something to solve the other direction.

edited Dec 09 '23 at 14:41

Anne Bauval

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asked Dec 09 '23 at 12:20

HHe

Is there a question here? – Another User Dec 09 '23 at 12:25
1

Did you prove one way? – Kavi Rama Murthy Dec 09 '23 at 12:25
Quick beginner guide for asking a well-received question + please avoid "no clue" questions: show some work and explain precisely what stops you. – Anne Bauval Dec 09 '23 at 14:29
Duplicate of https://math.stackexchange.com/questions/4202406 – Anne Bauval Dec 09 '23 at 16:54
There is a comment in the list of errata to the book stating that "This exercise is not as obvious as claimed in the beginning of the chapter. It deserves at least a hint." – Jochen Dec 10 '23 at 12:52

Relations between finite dimensional normed vector space and inner product space

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