From Vector Spaces to Periodic Functions

By Susam Pal on 30 Jan 2019

Vector Spaces

A fascinating result that appears in linear algebra is the fact that the set of real numbers \( \mathbb{R} \) is a vector space over the set of rational numbers \( \mathbb{Q}. \) This may appear surprising at first but it is easy to show that it is indeed so by checking that all eight axioms of vector spaces hold good:

  1. Commutativity of vector addition:
    \( x + y = y + x \) for all \( x, y \in \mathbb{R}. \)

  2. Associativity of vector addition:
    \( x + (y + z) = (x + y) + z \) for all \( x, y, z \in \mathbb{R}. \)

  3. Existence of additive identity vector:
    We have \( 0 \in \mathbb{R} \) such that \( x + 0 = x \) for all \( x \in \mathbb{R}. \)

  4. Existence of additive inverse vectors:
    There exists \( -x \in \mathbb{R} \) for all \( x \in \mathbb{R}. \)

  5. Associativity of scalar multiplication:
    \( a(bx) = (ab)x \) for all \( a, b \in \mathbb{Q} \) and for all \( x \in \mathbb{R}. \)

  6. Distributivity of scalar multiplication over vector addition:
    \( a(x + y) = ax + by \) for all \( a \in \mathbb{Q} \) and for all \( x, y \in \mathbb{R}. \)

  7. Distributivity of scalar multiplication over scalar addition:
    \( (a + b)x = ax + bx \) for all \( a, b \in \mathbb{Q} \) and for all \( x \in \mathbb{R}. \)

  8. Existence of scalar multiplicative identity:
    We have \( 1 \in \mathbb{Q} \) such that \( 1 \cdot x = x \) for all \( x \in \mathbb{R}. \)

This shows that the set of real numbers \( \mathbb{R} \) forms a vector space over the field of rational numbers \( \mathbb{Q}. \) Another quick way to arrive at this fact is to observe that \( \mathbb{Q} \subseteq \mathbb{R}, \) that is, \( \mathbb{Q} \) is a subfield of \( \mathbb{R}. \) Any field is a vector space over any of its subfields, so \( \mathbb{R} \) must be a vector space over \( \mathbb{Q}. \)

Problem

Here is an interesting problem related to vector spaces that I came across recently:

Define two periodic functions \( f \) and \( g \) from \( \mathbb{R} \) to \( \mathbb{R} \) such that their sum \( f + g \) is the identity function. The axiom of choice is allowed.

A function \( f \) is periodic if there exists \( p \gt 0 \) such that \( f(x + p) = f(x) \) for all \( x \) in the domain.

If you want to think about this problem, this is a good time to pause and think about it. There are spoilers ahead.

Solution

The axiom of choice is equivalent to the statement that every vector space has a basis. Since the set of real numbers \( \mathbb{R} \) is a vector space over the set of rational numbers \( \mathbb{Q}, \) there must be a basis \( \mathcal{H} \subseteq \mathbb{R} \) such that every real number \( x \) can be written uniquely as a finite linear combination of elements of \( \mathcal{H} \) with rational coefficients, that is, \[ x = \sum_{a \in \mathcal{H}} x_a a \] where each \( x_a \in \mathbb{Q} \) and \( \{ a \in \mathcal{H} \mid x_a \ne 0 \} \) is finite. The set \( \mathcal{H} \) is also known as the Hamel basis.

We know that \( b_a = 0 \) for distinct \( a, b \in \mathcal{H} \) because \( a \) and \( b \) are basis vectors.

In the above expansion of \( x, \) each \( x_a \) is a rational number that appears as the coefficient of the basis vector \( a. \) Therefore \( (x + y)_{a} = x_a + y_a \) for all \( x, y \in \mathbb{R}. \) Thus \( (x + b)_{a} = x_a + b_a = x_a + 0 = x_a. \) This shows that a function \( f(x) = x_a \) is a periodic function with period \( b \) for any \( b \in \mathcal{H} \setminus \{a\}. \)

Let us define two functions: \begin{align*} f(x) & = \sum_{a \in \mathcal{H} \setminus \{ b \}} x_a a, & g(x) & = x_b b. \end{align*} where \( b \in \mathcal{H} \) and \( x \in \mathbb{R}. \) Let us choose \( c \in \mathcal{H} \) such that \( c \ne b. \) Then \( f(x) \) is a periodic function with period \( b \) and \( g(x) \) is a periodic function with period \( c. \) Further, \[ f(x) + g(x) = \left( \sum_{a \in \mathcal{H} \setminus \{ b \}} x_a a \right) + x_b b = \sum_{a \in \mathcal{H}} x_a a = x. \] Thus \( f(x) \) and \( g(x) \) are two periodic functions such that their sum is the identity function.

References