Recently I started studying the connections between the algebraic structures and functional spaces.
I'm interested in the space of all continuous functions $C_p(X,Y)$ endowed with pointwise topology.I would like to know under which conditions this space has a structure of a group/ring/module or vector space. $C_p(X,Y)$=$\{f \in C(X,Y) , f : \text{is continuous} \}$.
For $C_p(X,Y)$ to be a topological group , the application $\alpha_1$ : $(f,g)$ $\to$ $f+g$
from $C_p(X,Y) \times C_p(X,Y)$ to $C_p(X,Y)$ and the application $\alpha_2$ : $f \to -f$ from $C_p(X,Y)$ to $C_p(X,Y)$ must be continuous.
I tried to find under what conditions $\alpha_1$ is continuous.
Let $U$ be an open set from $C_p(X,Y)$.i.e.
$U$=W($x_{1}$,$x_{2}$,...,$x_{n}$,$U_{1}$,$U_{2}$,...,$U_{n}$), where $x_{i} \in X$, $U_{i}$ are open sets in $Y$,$\forall i \in \{1,...,n\}$
$\alpha_1^{-1}(U)=\{(f,g) \in C_p(X,Y) \times C_p(X,Y) : (f+g)(x_i) \in U_{i}, \forall i.\}$
For $\alpha_1^{-1}(U)$ to be an open set of $C_p(X,Y)$ , $f(x_i)$ and $g(x_i)$ both must be in $U_i$ & for that $U_i$ must be a topological subgroup of the topological group $Y$?
Trying to identify with the results I found on some references regarding the space $C_p(X)$ ( where $Y=R$ ) , I think it depends on the codomain space $Y$ , if $Y$ is a topological group then $C_p(X,Y)$ is also a topological group.
I'm a little bit confused , Is that condition sufficent ? I think it depends also on the structure of the open sets $U_i$ , shouldn't they be subgroups of $Y$ to prove that $\alpha_1^{-1}(U)$ is an open set of $C_p(X,Y)$? Did I miss something here ?
It is pretty standard and well-known that $C_p(X,Y)$ is a topological group whenever $Y$ is; it's really just a subgroup of a product of copies of the group $Y$. Likewise, it's a real vector space when $Y$ is etc. Pointwise operations are continuous when those operation are continuous on the codomain.
And as $Y$ naturally embeds into $C_p(X,Y)$ via the constant functions (which form a subgroup/subspace) it's clear that having the wished for structure on $C_p(X,Y)$ will imply that $Y$ has to have that same structure, at least when we want that structure on $C_p(X,Y)$ to be connected to structure on $Y$..