What is Hilbert Space and Why is it Important in Mathematics?

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Discussion Overview

The discussion centers around the concept of Hilbert space, exploring its definition, properties, and significance in mathematics. Participants share their understanding of both finite and infinite-dimensional Hilbert spaces, as well as their preferences for learning resources.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants define a Hilbert space as a complete inner product space, while others emphasize its completeness as a pre-Hilbert space.
  • One participant describes finite-dimensional Hilbert spaces as equivalent to R^n, highlighting the use of the dot product and the concept of vector length.
  • Another participant explains that infinite-dimensional Hilbert spaces consist of certain infinite sequences of real numbers, with a focus on those sequences that have a finite squared length.
  • There is mention of generalizations of Hilbert spaces to higher dimensions, where length is defined by the finiteness of an integral.
  • Some participants express a preference for learning mathematics through traditional means rather than online resources, while others provide links to external sources for further reading.

Areas of Agreement / Disagreement

Participants express varying definitions and examples of Hilbert spaces, indicating that multiple competing views remain. There is no consensus on a singular definition or understanding of the concept.

Contextual Notes

Limitations include the potential ambiguity in definitions of completeness and the varying interpretations of dimensionality in Hilbert spaces. Some mathematical steps and assumptions regarding convergence and distance metrics remain unresolved.

Who May Find This Useful

This discussion may be useful for individuals interested in mathematical concepts, particularly those studying functional analysis or related fields in mathematics.

bhanukiran
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hilbert space??

hai,
what is hilbert space ?any important links known to you regarding that?please send some links .
 
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A Hilbert space is a complete preHilbert space...

I can't fill u in will links,because i don't like learning mathematics on the internet...

Daniel.
 
Or a hilbert space is a complete innerproduct space. All words defined by googling for them plus wolfram, or mathworld
 
a finite dimensional hilbert space is just R^n, whose points are finite sequences of numbers of length n, equipped with the usual dot product. Notice each vector has finite length, e.g. the squared length of (x,y) is x^2 + y^2.

a typical infinite dimensional hilbert space has as points certain infinite sequences of real numbers (x1,x2,...) but we want a dot product here too, so we set the squared length equal to the infinite sum x1^2 + x2^2 +...

of course here this infinite sequence may not have a finite sum. so we restrict attention to those sequences which do have a finite squared length. this subset of the space of all infinite sequences is a (separable) hilbert space.

so in an infinite dimensional euclidean space, most points have infinite distance from the roign, so we consider only those at finite distance from the origin. that's hilbert space.

there are then generalizations with higher (infinite) dimension as well, whose length is defined by some integral being finite.

e.g. take all functions on the unit interval whose square has finite integral. or all functions on the real line with that proeprty.


now that i reread matt's definit0on, of cousare the abstarct evrsion is that there is adot product,. hence defining a distance, and we require all cauchy sequences in this distance to converge. i hope my examples do this, i believe they do.
 
bhanukiran said:
hai,
what is hilbert space ?any important links known to you regarding that?please send some links .

http://mathworld.wolfram.com/HilbertSpace.html
it's got some finite-dimensional & infinite-dimensional examples
 
hello?? I already laid out those examples in considerably more detail than wolfram does.
 

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