I am really having trouble understanding this analogy, extending linear algebra to functional analysis. I fully understand the vector part, but for $v$ below I feel vague.
Here I list some notation and definitions of vectors in the text
$${\bf v\equiv {\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_N \end{bmatrix}}}\in\mathbb{R}^N\\{\bf v}: n\in \{1,2,3\dots N\}\rightarrow v_n\in\mathbb{R}$$
The main dierence (and analogy) between the Euclidean space and a vector space of functions is that the discrete integer index $n$ of the Euclidean vectors becomes a continuous index $\bf x$. Furthermore, we also let the value of the vector be complex
$$v:{\bf x}\in\mathbb{R}^N\rightarrow v({\bf x})\in\mathbb{C}$$
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Notice that we denote as $v$ the function, to be compared with the boldface notation $\bf v$ above, which denotes a vector; on the other hand we denote as $v({\bf x})$ the specific value of that function in $\bf x$, to be compared with the entry of the vector $v_n$
Instead let's start with familiar indexing set for $\mathbb{R}^3$ which is $I = \{x,y,z\}$ which correspond the $x,y$ and $z$ coordinates. Now consider all the possible functions $f: I \rightarrow \mathbb{R}$ typically called $\mathbb{R}^I$. Each of these functions assigns is fully described by three real numbers, namely $f(x),f(y)$ and $f(z)$. So here you can think of $f$ as a projection onto each coordinate since it allows us to recover the value of that coordinate. So when I write the vector as a tuple $(a,b,c)$ we can think of this as saying $f(x)=a,f(y)=b,f(z)=c$ and so there is this nice bijection between vectors in $\mathbb{R}^3$ that we're familiar with and the functions in $\mathbb{R}^I$. The reason we write $\mathbb{R}^3$ is because $\vert I \vert =3$ so you can sort of see where the notation comes from.
Now for functional analysis we generalize to a different indexing set. Let's use $I = [0,1]$ and again consider out functions $\mathbb{R}^I$. Now each number in $I$ is the name of a different coordinate. The fact that the indexing set is numbers isn't important as a vector space although the cardinality does make a big difference on things like dimension and the dual spaces. However our choice of $I$ gives us some nice analytical and topological properties to work with which is where the functional analysis comes in.
So in summary real valued functions over some indexing set are just labels for coordinates and if we pick the right indexing set we get to use calculus on them.