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sorry if I make some grammatical faults

When I read some maths text about inner product space and norms, I see that inner product spaces induces norms. But does it mean that the inner product space induce a norm or induce THE norm ? So is a norm on an inner product space unique ? Thanks !

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  • $\begingroup$ We says A norm because the space have also other norms, that they can not necessarily related to the inner product of the space. It is obvious that the norm induced by the inner product is unique because depend only by that inner product. An interesting question can be the following: if two norms are induced by two inner product and they are equal, then that inner products are equal? The answer is yes, because each inner product can be written in function of the norm induced on the space $\endgroup$ Jul 30, 2019 at 11:54
  • $\begingroup$ If you double a norm, you still get a norm. $\endgroup$ Jul 30, 2019 at 11:54
  • $\begingroup$ Thanks ! So if norms are different then inner product that induces these norms are different ? So if I have a space with an inner product define on it, then the norm induce by my inner product is unique ? So It's impossible to have two different norms on my space WHEN an inner product is define on it, am I wrong ? (whereas it is possible to have two different norms on a space without an inner product defined on) $\endgroup$
    – Cqfd
    Jul 30, 2019 at 14:13

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Each inner product determines a norm. If you know the norm and know that it comes from an inner product you can recover the inner product:

Finding an inner product given a norm

But a normed space can have different norms that come from different inner products. Consider $$ \langle (u,v) , (x,y) \rangle = ux + 2vy $$ in the plane.

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    $\begingroup$ and some norms don't come from any inner product, e.g., $\|(x,y)\|=|x|+|y|$ in the plane. $\endgroup$ Jul 30, 2019 at 12:09

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