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

Approximation Properties of Locally Convex Spaces and the Problem of Uniqueness of the Trace of a Linear Operator