Math 220 - April 7, 1999

The Vector Space of Functions on a Set

• Definition. If X is any set, then the set F(X) of all real-valued functions from X to the set R of real numbers is a vector space when the addition of functions and the multiplication of a function by a number is defined in a manner that is essentially the same as that used in the discussion of March 19.

• For each f in F(X) the set X_{f} of all points x in X for which f(x) <> 0 is called the support of f.

• If a is a given point of the set X the Kronecker delta function delta_{a} is the function defined by declaring delta_{a}(x) = 0 when x <> a and delta_{a}(a) = 1.

• Proposition. If F_{0}(X) denotes the set consisting of those functions in F(X) each of which, say f, has the property that f(x) = 0 except for finitely many values of x in X, then:

  1. For each a in X its Kronecker delta function delta_{a} is in F_{0}(X).

  2. F_{0}(X) consists of those functions in F(X) that have finite support.

  3. F_{0}(X) is a linear subspace of F(X).

  4. For each f in F_{0}(X) one has the formula

     f = SUM_{a}[ f(a) delta_{a} ] .

  5. The subset of F_{0}(X) consisting of all of the Kronecker delta functions delta_{a}, as a varies over all points of X, is a basis of F_{0}(X).

• In the case where X is the set R of all real numbers, other linear subspaces of F(R), aside from F_{0}(R), include:

  1. The set of all continuous functions.

  2. The set of all differentiable functions.

  3. The set of functions each of which is equal to the sum of its Taylor series at the origin.

  4. The set of linear functions.

Assignment for Friday, April 9

1. What is the dimension of the subspace of R^{n} consisting of all points (x_{1}, ..., x_{n}) such that

   x_{1} + 2 x_{2} + 3 x_{3} + ... + n x_{n} = 0 .

   Hint: Try the special cases n = 1, 2, and 3 first.

2. For n an integer, n >= 1, let F_{n} denote the the set {1, 2, 3, ..., n}. Find an invertible linear map from F(F_{n}) to R^{n}.

3. What can be said about the linear functions in F(R^{n}) that have finite support?

4. Prove the proposition stated above.

5. (Somewhat difficult) Let S be the vector space of infinite sequences. Define a map mu from S to S by

   mu(s)_{n} = {s_{1} + s_{2} + ... + s_{n}}/{n} .

   1. Verify that mu is linear.

   2. Compute the composition lim \circ mu \circ i , where i is the inclusion in S of the subspace S_{c} of convergent sequences.

6. (Difficult) Let X be any set, and let f_{1}, f_{2}, ..., f_{n} be elements of F(X). Show that f_{1}, f_{2}, ..., f_{n} are linearly independent if and only if there exist points x_{1}, x_{2}, ..., x_{n} in X such that the n \times n matrix ( f_{i}(x_{j})) is an invertible matrix.