The term 'closed' and also the term 'complete' are used to describe
vectors and spaces in quantum mechanics.
A set of vectors is 'closed' if a sequence of vectors converges to a
vector which is also a member of the set.
A space is complete if every Cauchy sequence of vectors converges to a
limit vector that is also in the space.
These two terms seem to say the same thing - it's pretty difficult for
me to be able to understand why they don't amount to the same thing -
they both refer to converging vectors.
Perhaps one way might be if someone could give an example of a set of
closed vectors that does not live in a complete space.
Or/Also a complete space that admits a set of vectors that isn't closed.
Thanks for any help.
David.


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David Macmanus wrote:

The term 'closed' and also the term 'complete' are used to describe
vectors and spaces in quantum mechanics.
A set of vectors is 'closed' if a sequence of vectors converges to a
vector which is also a member of the set.
A space is complete if every Cauchy sequence of vectors converges to a
limit vector that is also in the space.
These two terms seem to say the same thing - it's pretty difficult for
me to be able to understand why they don't amount to the same thing -
they both refer to converging vectors.
Perhaps one way might be if someone could give an example of a set of
closed vectors that does not live in a complete space.
Or/Also a complete space that admits a set of vectors that isn't closed.
Thanks for any help.
David.

If you're talking about subsets of a Hilbert space (or indeed any
complete metric space), then they are the same thing. Given a complete
metric space (X,d) then if A is a subset of X it's easy to show that A
is closed in X iff (A,d') is itself a complete metric space (where d'
is given by the restriction of d to A x A).

In general complete is an adjective that may, or may not, apply to any
metric space such as a normed space or an inner product space. A
complete normed space is called a Banach space and a complete inner
product space is called a Hilbert space.

In contrast it makes no sense to talk of a space being 'closed'. You
can only talk about a space being closed in (blah) - another space. If
X is a topological space then a subspace A is closed in X iff the
complement X - A is open in A. In the case that X is a metric space
(e.g. a normed/Hilbert space) then an alternative characterisation is
that A is closed in X iff whenever a_1,a_2,... is a sequence in A that
tends to a in X then necessarily a is in A. In the non-metric case you
need a generalisation of sequences (called nets) for this alternative
characterisation.

But the point, anyway, is that a space itself cannot be closed, it only
makes sense to talk about it being closed in another space. In contrast
completeness is an intrinsic property of the space itself.

David Macmanus wrote:

The term 'closed' and also the term 'complete' are used to describe
vectors and spaces in quantum mechanics.
A set of vectors is 'closed' if a sequence of vectors converges to a
vector which is also a member of the set.
A space is complete if every Cauchy sequence of vectors converges to a
limit vector that is also in the space.
These two terms seem to say the same thing - it's pretty difficult for
me to be able to understand why they don't amount to the same thing -
they both refer to converging vectors.
Perhaps one way might be if someone could give an example of a set of
closed vectors that does not live in a complete space.
Or/Also a complete space that admits a set of vectors that isn't closed.

If this is QM, we have a metric space.

(I) A metric space is closed if it contains all its limit points;
elements X s.t. for all eps there exists an element Y s.t.|X-Y| eps

(II) A metric space is complete if every Cauchy sequence is convergent,
i.e. for all sequences X_i s.t lim |X_j - X_k| = 0, j,k - oo.

There is an asymmetry between these conditions. (II) says that given
the indicated kind of sequence a limit exists and is in the space. (I)
says that _if_ a limit exists according to the definition, then it is
in the space.

If we start with a complete space and then drop in a limit point
according to definition (I), we can use that condition to construct a
Cauchy sequence converging on that point, which hence is in the space.
So II = I. But if we are given a space satisfying (I) and then drop
in a Cauchy sequence, however much we may be tempted to postulate a
limit to the sequence, demonstrate that the limit is in fact a limit
point in the sense of (I) and hence in the space, so that the Cauchy
sequences converges to a point in the space, we are apparently
unjustified in postulating the existence of a limit to the Cauchy
sequence either in or out of the space to begin with. I /= II

To make the two equivalent we would have to add that "All Cauchy
sequences converge to some point, whether or not in thespace containing
the sequence". In which case we could show that this limit was in
fact the limit point in sense (I), hence that closed spaces were
complete.

Possibly I'm reading too much into this, but that's a close reading of
the implications of the definitions gleaned from Mathworld.

"Edward Green" wrote in message
ups.com


(I) A metric space is closed if it contains all its limit points;
elements X s.t. for all eps there exists an element Y s.t.|X-Y| eps

Thanks.
What do s.t. and eps stand for?
Thanks.


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David Macmanus wrote:

"Edward Green" wrote in message
ups.com


(I) A metric space is closed if it contains all its limit points;
elements X s.t. for all eps there exists an element Y s.t.|X-Y| eps

Thanks.
What do s.t. and eps stand for?
Thanks.

"such that" and "epsilon"

You are welcome. However, I think you may probably safely ignore my
response and prefer michaeld's. I was about to add the following tag
to his post:

"Oops! That's what comes from trying to reason from the definitions
and not knowing the culture of the thing".

The idea that "closed" doesn't make much sense applied to a space would
tie in with the tacit assumption of the existence of the limit point,
independent of its existence in the target set, whereas in the
definition of "complete" it is said that the given sequence "converges"
implying both the existence and inclusion of the limit in the
set/space. The idea that the limit could exist but outside the target
set doesn't come up.

Thanks for posting a substantive question.

michaeld wrote:

[...]

In contrast it makes no sense to talk of a space being 'closed'. You
can only talk about a space being closed in (blah) - another space. If
X is a topological space then a subspace A is closed in X iff the
complement X - A is open in A.

Sorry, that should of course say 'open in X' not 'open in A'. A is
closed in X iff A^c := X-A is open in X.

[...]