Question on Hilbert spaces
Enrico Bernardini
I came across a strange passage in a book (*). This is about the
advantage of considering infinite sums of orthonormal vectors in
a Hilbert space, rather than finite sums of linearly independent
ones. Then it cites the example of L^2([a,b]), which "has
countable orthonormal bases consisting of simple funtions, while
each of its [finite?] bases is uncountable and we can only prove
that such a basis exists without being able to describe its
elements."
I can't make sense of this passage: how can a finite basis be
uncountable? What do the authors mean?
Thanks,
Enrico
(*) "Introduction to Hilbert spaces with applications" by
Lockenath Debnath and Piotr Mikusinski, page 99.
* Sent from RemarQ http://www.remarq.com The Internet's Discussion Network *
The fastest and easiest way to search and participate in Usenet - Free!
John Baez
Enrico Bernardini <e_bernardi...@hotmail.com.invalid> wrote:
>While studying the mathematical foundations of quantum mechanics,
>I came across a strange passage in a book (*). This is about the
>advantage of considering infinite sums of orthonormal vectors in
>a Hilbert space, rather than finite sums of linearly independent
>ones. Then it cites the example of L^2([a,b]), which "has
>countable orthonormal bases consisting of simple funtions, while
>each of its [finite?] bases is uncountable and we can only prove
>that such a basis exists without being able to describe its
>elements."
>
>I can't make sense of this passage: how can a finite basis be
>uncountable? What do the authors mean?
By a "finite basis" they mean a set S of linearly independent
vectors for which every vector in the space can be written as
a *finite* linear combination of vectors in S. As you note, the
term "finite basis" is inherently confusing, so I urge you to
use some other term, like "Hamel basis".
This concept of "Hamel basis" makes sense for any vector
space. On the other hand, an "orthonormal basis" only makes
sense for a Hilbert space. An "orthonormal basis" of a Hilbert
space is a set S of orthonormal vectors for which every vector
in the space can be written as a *limit* of finite linear
combinations of vectors in S.
What they are saying is that L^2([a,b]) has countable orthonormal
bases but only uncountable Hamel bases. When applying Hilbert
spaces to physics, the concept of Hamel basis hardly ever comes
up. The main reason for learning about it is to master the
distinction between finite linear combinations and limits of finite
linear combinations (also known as "infinite linear combinations").
In other words, it's good to get confused about this stuff, then
get unconfused, and then stop thinking about it most of the time.
The book has done an excellent job of getting you to the first stage.
Hein Roehrig
from the space can be written as a linear combination of *finitely*
many basis vectors.
A Hilbert space basis is a set of pairwise orthogonal vectors so that
each vector of the Hilbert space is the limit of a series sum x_k
where each x_k is from the Hilbert space basis.
In many cases, there are natural Hilbert space bases---since they only
approximate all vectors in the space, they are much "smaller" sets.
-Hein
Enrico Bernardini <e_bernardi...@hotmail.com.invalid> writes:
> [...] This is about the advantage of considering infinite sums of