Potpourri, 6

Let E be a nonempty finite set, and let V be a real or complex vector space. In these notes we write F(E, V ) for the vector space of V -valued functions on E. Suppose that E1, E2 are nonempty finite sets, and that V is the vector space of real or complex-valued functions on E2. In this case we can identify F(E1, V ) with the vector space of real or complex-valued functions on the Cartesian product E1 × E2, as appropriate. In any event F(E, V ) is basically the same as the tensor product of the vector space of real or complex-valued functions on E, as appropriate, with V . Of course the tensor product of two vector spaces, both real or both complex, is symmetric up to canonical isomorphism in the two vector spaces. With F(E, V ) we may unravel this symmetry a bit, depending on the circumstances. For each x ∈ E, let δx(y) denote the scalar-valued function on E which is equal to 1 when y = x and to 0 when y 6= x. These functions form a basis for the scalar-valued functions on E, and in effect we may treat the vector space of scalar-valued functions on E as having a distinguished basis in this way. We can also think of the scalar-valued functions on E as being a commutative algebra over the real or complex numbers, as appropriate, using ordinary pointwise multiplication as the product in the algebra. We can think of F(E, V ) as being a module over the commutative algebra of scalar-valued functions, since we can multiply a vector-valued function by a scalar-valued function. Suppose that A is any linear transformation on V . There is a unique linear transformation à on F(E, V ) such that Ã(f)(x) = A(f(x)) for every V -valued function f on E and every x ∈ E. Alternatively, à is characterized by the property that Ã(f v) = f A(v) for every scalar-valued function f on E and every vector v ∈ V . Now suppose that T is any linear transformation acting on F(E,R) or F(E,C), as appropriate. There is a unique linear transformation T̂ on F(E, V ) such that T̂ (f v) = T (f) v for all scalar-valued functions f on E and all v ∈ V . Thus one can also think of F(E, V ) as modules over the algebras of linear transformations acting on the vector space of scalar-valued functions on E and on V . The actions of à and T̂ on F(E, V ) described in the preceding paragraphs automatically commute, by construction. Let us focus for the moment on scalar-valued functions. Let E be a nonempty