### 3.4 Function spaces

A connection 1-form can be reconstructed uniquely if all its holonomies are known [172]. It is thus
sufficient to parameterize the configuration space by matrix elements of for all edges in space. This
defines an algebra of functions on the infinite-dimensional space of connections , which are multiplied as
-valued functions. Moreover, there is a duality operation by complex conjugation, and if the structure
group is compact a supremum norm exists, since matrix elements of holonomies are then bounded.
Thus, matrix elements form an Abelian -algebra with unit as a sub-algebra of all continuous functions
on .
Any Abelian -algebra with unit can be represented as the algebra of all continuous functions on a
compact space . The intuitive idea is that the original space , which has many more
continuous functions, is enlarged by adding new points to it. This increases the number of continuity
conditions and thus shrinks the set of continuous functions. This is done until only matrix
elements of holonomies survive when continuity is imposed, and it follows from general results that
the enlarged space must be compact for an Abelian unital -algebra. We thus obtain a
compactification , the space of generalized connections [24], which densely contains the space
.

There is a natural diffeomorphism invariant measure on , the Ashtekar–Lewandowski
measure [20], which defines the Hilbert space of square integrable functions on the
space of generalized connections. A dense subset of functions is given by cylindrical functions
, which depend on the connection through a finite but arbitrary number of holonomies.
These cylindrical functions are associated with graphs formed by the edges , …, . For
functions cylindrical with respect to two identical graphs the inner product can be written as

with the Haar measure on . The importance of generalized connections can be seen from the fact
that the space of smooth connections is a subset of measure zero in [224].
With the dense subset of we obtain the Gel’fand triple

with the dual of linear functionals from to the set of complex numbers. Elements of
are distributions, and there is no inner product on the full space. However, one can define
inner products on certain subspaces defined by the physical context. Often, those subspaces
appear when constraints with continuous spectra are solved following the Dirac procedure. Other
examples include the definition of semiclassical or, as we will use it in Section 7, symmetric
states.