The trace on the n×n matrices is the unique linear functional satisfying the conditions (1) and (2) .

For matrices (the set of m×m matrices) and , the functional is bilinear, and therefore extends to a linear functional on . So we just have to check that f satisfies (1) and (2). Condition (1) is bilinear, so you only need to check it on simple tensors:

Condition (2) is trivially satisfied, so f is the functional Tr on the mn×mn matrices.