Hilbert Space

Date: 02/04/98 at 00:24:59
From: Renee Anderson
Subject: Hilbert Space

I am an eleventh grade high school student, and I'm doing reseach on
quantum computations. Could you please tell me what Hilbert Space is?

Date: 02/04/98 at 16:21:48
From: Doctor Rob
Subject: Re: Hilbert Space

See the following URL on the World Wide Web:

  http://www.astro.virginia.edu/~eww6n/math/HilbertSpace.html   

It is a vector space over a field with an inner product, such that the
inner product yields a norm turning the vector space into a metric 
space. That is a terribly technical definition, which you may not 
understand.

Over the field of real numbers, the set of real n-tuples with 
componentwise addition and the usual dot-product forms a Hilbert 
space. The norm is the square root of the sum of the squares of the 
components of the n-tuple. The norm of the difference between two n-
tuples is then the distance between them, and it satisfies the usual 
properties of a distance: it is never negative, it is zero if and only 
if the two n-tuples are the same, and it satisfies the triangle 
inequality.  (In fact, it is the usual Euclidean distance.) This 
distance function makes the set a metric space.

Over the complex numbers, the set of complex n-tuples with 
componentwise addition and the inner product <u,v> = u . conjugate(v) 
forms a Hilbert space.  The norm is ||v|| = Sqrt[<v,v>] = square root 
of the sum of the squares of the absolute values of the components 
of v. The norm of the difference between two n-tuples is then the 
distance between them, and it satisfies the usual properties of a 
distance:  it is never negative, it is zero if and only if the two 
n-tuples are the same, and it satisfies the triangle inequality. This 
distance function makes the set a metric space.

The above Hilbert spaces have dimension n. Some Hilbert spaces can be
infinite-dimensional.

In quantum computation, the metric spaces are of the second type 
described above, and the dimension is usually 2^k, when there are k 
quantum bits.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/