Function space in QM I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to me. I've also read this introduction but found it too short to really grasp the concept.
I've been recommended Cohen-Tannoudji's QM book. Do you have any other recommendations or explanations to offer?
My goal is not only to understand the concept, but use it. Also, for the interested, I'm a chemistry major - mathematics is poorly taught to chemistry majors, in my experience.
 A: If your working definition of "vector" is "an ordered list of numbers", then you can start there.
Let's consider $v=(A,B,C)$ where $A,B,C$ are all real numbers. This gadget would be a vector for us. How could one look upon it as a function? Well we could name the positions #1, #2, and #3, and say $v(1)=A, v(2)=B, v(3)=C$. Therefore the domain (things for $i$ in $v(i)$) is $1,2,3$ and the output is determined by whatever is in that position.
Now let's think of a sequence of real numbers $a_1, a_2,\dots$. If I dress them up: $v=(a_1,a_2,\dots)$ this is also a (very long) vector. To think of it as a sequence we can pull the same trick of using the $i$'th position to record the output for $i$, and so $v(i)=a_i$ has given you a function from $\mathbb{N}$ into $\mathbb{R}$. Really, $a_i$ was already in functional notation, and you probably wouldn't need to introduce $v(i)$, but I'm making a point here :)
But why should we stop with a finite set or $\mathbb{N}$? Could we use $\mathbb{R}$ as subscripts for a vector? Sure, except now that the inputs for $i$ are not discrete anymore, it becomes impossible for us to physically write a vector with continuum many positions. But we can still imagine it. Just imagine you have a bunch of positions, each labeled with a real number $r$, and the whole bunch are bracketed by parentheses (). In the $r$'th spot, you put the output of $v$, and so we would write $v(r)=Q$, where $Q$ is that particular output.
Really, order isn't even necessary. Given any function from a set $X$ into a set $Y$ you just imagine a bunch of positions, one for each thing in $X$, surrounded by parentheses. In each position you are allowed to fill in a thing from $Y$. Let's call the whole vector $f$. What is $f$'s value at $x\in X$? Well, whatever value of $y$ is in the $x$th position of course! We denote this with $f(x)=y$. 
In this way $f$ can be thought of as a very long vector with entries indexed by $X$, and each entry contains the value of $Y$ which is the output.
