- #1

WWGD

Science Advisor

Gold Member

2019 Award

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## Main Question or Discussion Point

Hi All,

AFAIK, the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that , in H, the norm is generated by an inner-product, i.e., for every vector ##v##, we have## ||v||_H= <v,v>_H^{1/2} ##, and a generalized version of the Pythagorean theorem is satisfied (Polarization Identity, Parallelogram Law). Why is this so special, and, what is it this property gives rise to that is not the case with more general inner-product normed vector spaces?

AFAIK, the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that , in H, the norm is generated by an inner-product, i.e., for every vector ##v##, we have## ||v||_H= <v,v>_H^{1/2} ##, and a generalized version of the Pythagorean theorem is satisfied (Polarization Identity, Parallelogram Law). Why is this so special, and, what is it this property gives rise to that is not the case with more general inner-product normed vector spaces?

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