I'm working my way through Axler's "Linear Algebra Done Right" (3rd ed.), and I'm getting stuck on section 1.23, which says:
- If $S$ is a set, then $\textbf{F}^S$ denotes the set of functions from $S$ to $\textbf{F}$.
- For $f, g \in \textbf{F}^S$, the sum $f + g \in \textbf{F}^S$ is the function defined by $$(f + g)(x) = f(x) + g(x)$$ for all $x \in S$.
- For $\lambda \in \textbf{F}$ and $f \in \textbf{F}^S$, the product $\lambda f \in \textbf{F}^S$ is the function defined by $$(\lambda f)(x) = \lambda f(x)$$ for all $x \in S$.
As an example of the notation above, if $S$ is the interval [0,1] and $\textbf{F} = \textbf{R}$, then $\textbf{R}^{[0,1]}$ is the set of real-valued functions on the interval [0,1].
In the next paragraph, the author goes on to assert the following:
Our previous examples of vector spaces, $\textbf{F}^n$ and $\textbf{F}^\infty$, are special cases of the vector space $\textbf{F}^S$ because a list of length $n$ of numbers in $\textbf{F}$ can be thought of as a function from {1, 2, ..., $n$} to $\textbf{F}$ and a sequence of numbers in $\textbf{F}$ can be thought of as a function from the set of positive integers to $\textbf{F}$. In other words, we can think of $\textbf{F}^n$ as $\textbf{F}^{\{1,2,...,n\}}$ and we can think of $\textbf{F}^\infty$ as $\textbf{F}^{\{1,2,...\}}$.
It's at this point I get confused, due mostly to the example which relies on $\textbf{R}^{[0,1]}$. It seems to me that then number of elements in $\textbf{R}^{[0,1]}$ should be uncountably infinite and that I should be able to generate any value between $-\infty$ and $\infty$ from [0,1] using some member of $\textbf{R}^{[0,1]}$, which feels a whole lot like generating a point in $\textbf{R}^\infty$.
If that's the case, then what's the difference between a tuple generated from $\textbf{R}^{\{1,2,...,n\}}$ and one generated from $\textbf{R}^{\{1,2,...,n+1\}}$?
I think my difficulty lies in not understanding implicit restrictions on the notation. The answer to this specific sub-question may be a shortcut to understanding: Suppose I'm trying to think of a particular point in $(x,y,z)\epsilon\textbf{R}^3$ as $\textbf{R}^{\{1,2,3\}}$ where $f,g,h\epsilon\textbf{R}^{\{1,2,3\}}$. Must I think of that point in $\textbf{R}^3$ as $(f(1)=x, f(2)=y, f(3)=z)$, or can I think of it as $(f(1)=x, g(2)=y, h(3)=z)$?
