These structures are the objects in the representative example of a compact closed category. Modules are analogous to these structures defined over rings, and an algebra equips one of (15[1])these structures with a bilinear product. A representation is a group homomorphism from a group G to the automorphism group of one of these structures. The set of linear (15[1])functionals on (15[1])one of these structures forms [emphasize] another one of these structures called the (*) dual (15[1])space. Norms assign elements of these (10[1])structures (10[2])to non-negative reals (-5[1])and may be induced by an inner product. (10[1])Linear maps are functions (10[1])between these structures, which are (-5[1])closed (10[1])under addition and scalar multiplication. The number of elements of the basis of one of (10[1])these (10[1])structures gives (10[1])their dimension. For 10 points, identify these sets consisting of objects that may have a magnitude and (10[1]-5[1])direction. ■END■ (10[2])