Whats the difference between hilbert and pre-hilbert space?

[bilalkadri]

Hilbert space

What is the difference between hilbert space and a pre hilbert space.What do we mean by "if the series are all convergent" then a hilbert and a pre hilbert space are same???? any clue

 

[the_jackal]

Re: Hilbert space

A Hilbert Space is a complete preHilbert
space. That is, ||v||= (v, v)½ . Not hard to prove, I omit it.

Any pre-Hilbert space that is complete is called a Hilbert space H . Complete means
that for any Cauchy sequence xn that belongs to H there exists a vector x belongs H such that xn tends to x in H,
i.e.,
||x - xn|| = 0 as n tends to infinity.\]

 

[mayyan]

Re: Hilbert space

about the defenition of a complete space as far as i know it say that any chushy sequence in the space converge inside the space.

[ from some N and n,m>N |Xm-Xn| --> 0 ]

 

[dazhen]

Re: Hilbert space

mayyan,

I think you're right about the definition of completeness. But what you wrote at last is just the mathematical description of convergence of Couchy sequence. The mathematical words on completeness should be,

[Any Couchy sequence Xn in space H, for some N, n>N, |Xn-X|-->0, X \[\in\] H].

 

[a_aziz]

Hilbert space

hi
I don't know what is pre hilbert space.
please give me more datails about this.