Hilbert space?

  • Thread starter bhanukiran
  • Start date
hilbert space??

hai,
what is hilbert space ?any important links known to you regarding that?please send some links .
 

dextercioby

Science Advisor
Homework Helper
Insights Author
12,951
533
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... :grumpy:

Daniel.
 

matt grime

Science Advisor
Homework Helper
9,394
3
Or a hilbert space is a complete innerproduct space. All words defined by googling for them plus wolfram, or mathworld
 

mathwonk

Science Advisor
Homework Helper
10,733
912
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. thats 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.
 

mathwonk

Science Advisor
Homework Helper
10,733
912
hello?? I already laid out those examples in considerably more detail than wolfram does.
 

Related Threads for: Hilbert space?

  • Posted
Replies
1
Views
423
  • Posted
Replies
15
Views
4K
Replies
35
Views
1K
Replies
11
Views
4K
  • Posted
2
Replies
30
Views
3K
Replies
2
Views
3K
  • Posted
Replies
5
Views
5K
Replies
3
Views
603

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top