These structures are the objects in the representative example of a compact closed category. Modules are analogous to these structures (15[1])defined (15[1])over (-5[1])rings, (15[3]-5[1])and an algebra equips (15[1])one of 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 functionals on one of these structures forms [emphasize] another one of these structures called the (*) dual space. (10[1])Norms assign elements of these structures to non-negative reals and may be induced by an inner product. Linear (-5[1])maps (10[1])are functions (-5[1])between these structures, which are closed under addition and scalar multiplication. The number of elements of the basis of one of these structures (10[1])gives their dimension. (-5[1])For 10 (-5[1])points, identify these sets consisting of objects that may have a magnitude and direction. (10[2])■END■ (10[2]0[2])