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.