Parallelogram Equality+Inner Product Spaces

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    Parallelogram Product
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SUMMARY

The discussion confirms that a norm satisfying the parallelogram equality must originate from an inner product, specifically through the use of the polarization identity. To demonstrate this, one must understand the parallelogram law and apply the polarization identity: \| x+y\|^2 - \|x-y\|^2 = 4. The proof involves defining an inner product using this identity in a normed linear space where the parallelogram law holds, establishing that the space is indeed an inner product space. Additionally, examples such as l^\infty, l^1, and C[a,b] with the uniform norm are provided to illustrate spaces that do not satisfy the parallelogram law.

PREREQUISITES
  • Understanding of the parallelogram law in normed spaces
  • Familiarity with the polarization identity
  • Knowledge of inner product spaces and their properties
  • Basic concepts of normed linear spaces
NEXT STEPS
  • Study the proof of the polarization identity in detail
  • Explore the properties of inner product spaces and their applications
  • Investigate examples of normed spaces that do not satisfy the parallelogram law
  • Learn about the implications of fields of characteristic 2 in linear algebra
USEFUL FOR

Mathematicians, students of functional analysis, and anyone interested in the properties of normed linear spaces and inner product spaces.

gravenewworld
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It is true that if a norm satisfies the parallelogram equality then it must come from an inner product right (i.e. < , > is an inner product).? How in the world could you go about proving/showing this?
 
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ya that's true; here's how it's done:
a) you need to know the parallelogram law (duh) but also the polarization identity: \| x+y\|^2 - \|x-y\|^2 = 4&lt;x,y&gt;

b) let V be a normed linear space in which the parallelgram law holds. Define <x,y> by the polarisation identity & prove that V with that inner product is an inner product space, and that \|x\| = \sqrt{&lt;x,x&gt;}

b*) see that spaces like l^\infty, l^1, C[a,b] with the uniform norm, c_0, don't satisfy the parallelogram law, and that there's no inner product (by a) ) that gives the norms for those spaces
 
You are of course not allowing fields of characteristic 2.
 

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