Preliminaries

The prerequisite for a thorough understanding of this paper is familiarity with the basics of separable Banach space theory as developed in . This material has been presented in several places: [3, §§2-5], [4, §1], [12, §4], [20, § II.10]. We briefly review some of the concepts that we shall need.

Within
,
a (code for a) *complete separable metric space*
is defined to be a countable set
together with a function
satisfying *d*(*a*,*a*)=0,
*d*(*a*,*b*)=*d*(*b*,*a*), and
.
A (code for a)
*point* of *X* is defined to be a sequence
of elements of *A* such that
.
We extend *d* from *A* to *X* in the
obvious way. For
we define *x*=*y* to mean that *d*(*x*,*y*)=0.

Within
,
(a code for) an *open set* in *X* is defined to
be a sequence of ordered pairs
where
and
,
the rational numbers. We write to mean that
*d*(*a*_{m},*x*)<*r*_{m} for some
.
A *closed set*
is defined to be the complement of an open set *U*,
*i.e.*,
.

It will sometimes be necessary to consider a slightly different
notion. A (code for a) *separably closed set*
is defined to be a countable sequence of
points
.
We write
to mean that for all
there exists
such that
.
It is provable in
(but not in weaker systems) that for every
separably closed set *K* there exists an equivalent closed set *C*,
*i.e.*,
.
For
further details on separably closed sets, see
[2,3,4].

Within
,
a *compact set*
is defined to be a
separably closed set such that there exists a sequence of finite
sequences of points
,
,
,
such that
for all
and all
there exists
with
*d*(*x*,*x*_{ni})<1/2^{n}. The sequence of positive integers *k*_{n},
,
is also required to exist. It is provable in
that
compact sets are closed and located [8]. It is provable in
that compact sets have the Heine-Borel covering property,
*i.e.*, any covering of *K* by a sequence of open sets has a
finite subcovering.

Within
,
a (code for a) *separable Banach space*
is defined to be a countable pseudonormed vector space
*A* over
**Q**. With
,
*X* is a complete separable
metric space and has the usual structure of a Banach space over
**R**.
A *bounded linear functional*
may be defined as a
continuous function which is linear. The equivalence of continuity
and boundedness is provable in
.
We write
to
mean that
for all .