First off: I haven't taken a functional analysis course yet, so please keep that in mind when explaining. I just rather randomly read about this stuff and it started to interest me.
I'm having trouble understanding how the following contorted mappings work. I don't want any more details than necessary; just the plane explanation which functions they take to which space and why it makes sense to define the mapping in the way they are defined (maybe like a diagram in words) and so on.
1) In this Wikipedia article it was mentioned, that there is a natural map $F$ from a vector space $V$ over a field $F$ to V'' (the dual of the dual, meaning the space of all linear functionals that take themselves linear functionals as arguments and return values from $F$, if I understood it correctly ), by $F(x)(f)=f(x)$.
(The notation $F(x)(f)$ would imply that $F(x)$ is itself a function taking as an argument another function $f$. But what would be the domain and range of this $f$ ? And why does it make sense to assign the value of $F(x)(f)$ the value of $f$ at $x$)
2) Similar question (maybe more difficult), to be found in these lecture notes,page 61 bottom: Let $I$ be compact and $E,F$ Banach spaces and consider the mapping $\mathscr{F}:C(I,L(E,F)) \rightarrow L(E,C(I,F))$ defined by $\mathscr{F}(f)=(x\mapsto (t\mapsto f(t)(x)))$. Now I understand that to $f$ there is a map associated, lets call it $\mathscr{G}:E \rightarrow C(I,F)$, such that $\mathscr{G} (x)$ is a continuous map from $I$ to $F$, $t\mapsto f(t)(x)$. But what I don't understand is, why the author says this mapping simulates the change of variables.If it did, why did the constraints to work in the space of continuous respectively linear maps had to be imposed? And does why this (latest) mapping from $I$ to $F$ takes the value $f(t)(x)$ ?