Proving that given this property then the norm is induced by a inner product

Asked Nov 05 '15 at 13:48

Active Nov 05 '15 at 13:48

Viewed 49 times

Let $(X,||\cdot||)$ be a normed space such that, for $x,y\in X$

$$||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2$$.

Then I want to check that $||\cdot||$ is induced by an inner product, so what I was trying to do is to check that the norm looks like $\sqrt{(x,x)}$ but I can't fugure out how to do this only knowing the above equation, then I have tried to use the polarization equalities to get the result, the thing is that I want to check that the norm is induced by a inner product and not the converse.

Then, Can someone help me to prove this result please?

Thanks in advance.

asked Nov 05 '15 at 13:48

user162343

3,245

It seems that you're on the right track: The polarization identity tells you what the inner product ought to be, so one need only show that for $||\cdot||$, the operator $X \times X \to \Bbb R$ defined by the usual polarization identity is an inner product, and that the norm it induces is $||\cdot||$ itself. – Travis Willse Nov 05 '15 at 13:50
Can you clarify the last part of your comment please? :) – user162343 Nov 05 '15 at 13:52
the thing is How do you proof the last part ? – user162343 Nov 05 '15 at 13:57
1

If we denote the product defined by the polarization formula by $\langle,\cdot, ,,\cdot,\rangle$, then the second step is simply checking that $\langle x, x\rangle = ||x||^2$ – Travis Willse Nov 05 '15 at 14:00
Right jajajaja I apologize for the question jajajaja – user162343 Nov 05 '15 at 14:02
@Travis But verifying that $\langle\cdot,\cdot\rangle$ actually is an inner product is not at all trivial. – David C. Ullrich Nov 05 '15 at 14:07
So Is the second answer of this http://math.stackexchange.com/questions/21792/norms-induced-by-inner-products-and-the-parallelogram-law complete? :) – user162343 Nov 05 '15 at 14:11
@DavidC.Ullrich Yes, I certainly did not intend to suggest otherwise. – Travis Willse Nov 05 '15 at 14:17
@user162343 In what sense is this question not a duplicate of the linked one? – Travis Willse Nov 05 '15 at 14:19
So then what do you think about my above comment ? :) – user162343 Nov 05 '15 at 14:19
@Travis Ok. Sounded to me like you were saying the problem was trivial, which it's really not. – David C. Ullrich Nov 05 '15 at 14:20
Is because I have seen that question whaen I post mine in the right hand side od my computer screen :) sorry, but there are things of continous functions that are not expalined right? – user162343 Nov 05 '15 at 14:21

Proving that given this property then the norm is induced by a inner product

0 Answers0