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.