Vector Spaces as Sets of Functions

Vector spaces can be more abstract instead of our typical conceptualization. Our next example of a vector space involves a set of functions.

${\mathbb{F}}^S$

We define ${\mathbb{F}}^S$ with the following:

• If $S$ is a set, then ${\mathbb{F}}^S$ denotes the set of functions from $S$ to $\mathbb{F}$. In other words, each function in ${\mathbb{F}}^S$ takes an input from $S$ and gives an output in the field $\mathbb{F}$.
• For $f,g\in {\mathbb{F}}^S$, the sum $f+g\in {\mathbb{F}}^S$ is the function defined by

$$(f+g)(x)=f(x)+g(x)$$

• For all $\lambda \in \mathbb{F}$ and $f\in {\mathbb{F}}^S$, the product $\lambda \mathbb{F}\in {\mathbb{F}}^S$ is the function defined by

$$(\lambda f)(x)=\lambda f(x)$$

for all $x\in S$.

This perspective generalizes the concept of vectors. While we usually think of vectors as ordered tuples (like points in ${\mathbb{R}}^n$), in this more abstract setting, we treat entire functions as vectors.

As an example of the notation above, if $S$ is the interval $[0,1]$ and $\mathbb{F}=\mathbb{R}$, then ${\mathbb{R}}^{[0,1]}$ is the set of real-valued functions on the interval $[0,1]$. This means the the domain of each function are $[0,1]$, which is mapped to a real value $\mathbb{R}$.

• The elements of the vector space ${\mathbb{R}}^{[0,1]}$ are real-valued functions on $[0,1]$, not lists. In general, a vector space is an abstract entity whose elements might be lists, functions, or weird objects.

We can show that ${\mathbb{F}}^S$ is a vector space by considering the following:

• If $S$ is a nonempty set, then ${\mathbb{F}}^S$ (with the operations of addition and scalar multiplication as defined above) is a vector space over $\mathbb{F}$.
• The additive identity of ${\mathbb{F}}^S$ is the function $0:S\to \mathbb{F}$ for all $x\in S$, defined by:

$$0(x)=0$$

for all $x\in S$.

• For all $f\in {\mathbb{F}}^S$, the additive inverse of $f$ is the function $-f:S\to F$ defined by:

$$(-f)(x)=-f(x)$$

for all $x\in S$.

The vector space ${\mathbb{F}}^n$ is a special case of the vector space ${\mathbb{F}}^S$ because each $(x_1,\dots ,x_n)\in {\mathbb{F}}^n$ can be thought of as a function $x$ from the set $\{1,2,\dots ,n\}$ to $\mathbb{F}$. We just write $x(k)$ instead of writing $x_k$ for the $k$-th coordinate of $(x_1,\dots ,x_n)$.

• In other words, we can think of ${\mathbb{F}}^n$ as ${\mathbb{F}}^{\{1,2,\dots ,n\}}$.
• Similarly, we can think of ${\mathbb{F}}^{\infty }$ as ${\mathbb{F}}^{\{1,2,\dots \}}$.