  In the present article, it is proved that every nuclear operator in a locally
convex space E has a well-defined trace if E possesses the approximation
property. However, even if a space possesses the approximation property this
still does not guarantee a positive solution of A. Grothendieck's uniqueness
problem for this space. Below, we present an example of a quasi-complete space
with the approximation property in which it is not possible to define the trace
for all Fredholm operators (in the sense of A. Grothendieck). We prove that the
uniqueness problem has a positive solution if E possesses the "bounded
approximation property." Preliminary information and results are presented in
Section 2. A number of approximation-type properties of locally convex spaces
and relations between these properties are considered in Section 3. The
principal results of the present study, along with certain corollaries from
these results (for example, the existence of a matrix trace), may be found in
Section 4.
