Let $H$ be a separable complex Hilbert space. If $\dim(H)<\infty$, then the trace of every operator $T$ is well defined, and it can be computed from the matrix of $T$ in any basis. Although this definition can be shown to be basis independent, it still refers to some choice of basis, and must be followed by a proof of well definition. On the other hand, there are also several equivalent definitions of the trace (in finite dimensions) that do not refer to a basis. (E.g., the sum of eigenvalues repeated according to multiplicity; or via the exterior algebra; or via the isomorphism of $H\otimes H^*$ with $\operatorname{End}H$.)
If $\dim H=\infty$, the situation is more complicated, but, given an orthonormal basis, we can still define the trace in the analogous way as before for a subset of operators called trace class operators. But I have never seen an intrinsic (basis-free) version of these notions in infinite dimensions. Can the notions of trace class operators and trace in infinite dimensions be characterized intrinsically, i.e., without reference to any basis?