scitposting 3: functions are so cool

NO. Not the high school treatment of functions where they just find the zeroes of boring quadratic functions and the period of simple trigonometric functions. I’m talking about the vector-interpretation of functions. You got that right, functions are vectors. Not the usual real-valued vectors in 2-space or 3-space or any general n-dimensional space – you know, they one you could write as an n-tuple (x₁,…,x_n). I’m talking about continuous functions, which are INFINITE DIMENSIONAL VECTORS. Ok, so maybe I should define what vectors are first.

Vectors are elements in a vector space. That’s it, no comment on how many dimensions it should have, or whether or not it can be represented as as (x₁,…,x_n). But what exactly is a vector space?

Definition. A set V is said to be a vector space such that for any elements u, v, and w in V and for a scalar field F with a and b in F, there exists operations + and · such that

1. u + v is in V.
2. u + v = v + u
3. (u + v) + w = u + (v + w)
4. There exists an additive identity 0 such that v + 0 = 0 + v = v.
5. There exists an additive inverse -v such that v + (-v) = (-v) + v = 0.
6. a·v is in V.
7. a·(u + v) = a·u + a·v
8. (a+b)·v = a·v + b·v
9. a·(b·v) = (a·b)·v
10. There exists a multiplicative identity 1 in V such that 1·v = v.

This makes sense for vectors in Euclidean space – you know, the one with the usual point-wise addition, and real (or complex) valued scalar multiplication. But this definition also applies for functions. Let C(R) be the set of continuous functions over the real numbers R, let f, g, and h be in C(R), and let a and b be complex numbers.

1. f + g is also continuous therefore also in C(R)
2. f + g = g + f
3. (f + g) + h = f + (g + h)
4. If z(x) = 0, then f + z = z + f = f. z is the zero constant function.
5. f + (-f) = (-f) + f = 0
6. af is also a continuous function therefore also in C(R)
7. a(f + g) = af + ag
8. (a + b)f = af + bf
9. a(bf) = (ab)f
10. 1f = f

See? They fulfill the properties of a vector space. So C(R) is a vector space!! BUT, there is a key difference: C(R) is infinite dimensional!

What do we mean when we say “dimension” when we talk about vector spaces? Intuitively, dimension in Euclidean space is the number of perpendicular lines or axes you can draw intersecting at a point. In 2-space, you can only draw 2 (x and y axes), and in 3-space, you can draw 3 (x, y, and z). These axes can be represented by tuples, so for two space the positive direction of these axes are represented by the unit vectors {(1,0), (0,1)} and for 3-space, {(1,0,0), (0,1,0), (0,0,1)}. These basis vectors cannot be written as a scalar multiple of the other, hence they are said to be linearly independent . All the other vectors in these spaces can be represented as a linear combination of these unit vectors (like, a scalar multiple of one of the basis vectors plus other scalar multiples of the others). For example, (2,5,3) = 2(1,0,0) + 5(0,1,0) + 3(0,0,1), (1,2,3) = (1,0,0) + 2(0,1,0) + 3(0,0,1), and generally (x,y,z) = x(1,0,0) + y(0,1,0) + z(0,0,1). This generalizes to higher dimesions 4, 5, 6, and so on.

Mathematicians define the dimension of the vector space as the minimum number of linearly independent vectors that can span the entire vector space. So what is the minimum number of linearly independent functions that can span the entire set of continuous functions over the reals. Well there isn’t any. You need infinitely many functions.

Why? Take the set of polynomials, which is a subset of the set of continuous functions over the real numbers C(R). Any polynomial can be expressed as a linear combination of {1, x, x², x³, x⁴,…}. Already there is an infinite number of elements in this set. So the set of polynomials is infinite dimensional.

This can only mean that C(R) is infinite dimensional because a finite dimensional space cannot contain and infinite dimensional set. And we haven’t even considered other continuous functions: e^kt, sin(nx), and cos(mx), where k, n, and m are real numbers.

Actually the study of infinite dimensional vector space is already a field of research itself, called functional analysis. That’s why functions are so cool.