I have a problem with the notation of function spaces. I append the example where my understanding of the functions space is presented. Please correct me if I did any mistake (and confirm if I did correctly).
Let $f_1,\dots,f_p : \mathbb{R}^N \rightarrow \mathbb{R}$ be continuous functions. Let define a set of functions $A$ that includes previously defined functions $f_1,\dots,f_p$.
$$ A = \{ f_i, i\in\{1,\dots,p\} \} $$
Then I want to define function space $C$ defined with functions in the set $A$ as dimensions. In space $C$ I can now define vector $\mathbf{c} = [c_1, \dots, c_p]$, which I want it to represent function $f(\mathbf{c}, \mathbf{x})$ with N-dimensional input vector $\mathbf{x} = [x_1, \dots,x_N]$
$$ f(\mathbf{c}, \mathbf{x}) = \sum_{i=1}^p c_i f_i(\mathbf{x}) $$
---- From this point on my plan is to define a probability distribution over functions space $C$ (each vector in $C$, which is a weighted sum of functions, will have user-defined probability) and do different operations on that. -----
So the question:
- Do I need to introduce any other concept?
- Have I used the correct notation for: a set of functions $A$, function space $C$, vector in function space $\mathbf{c}$, a function $f(\mathbf{c}, \mathbf{x})$ that is represented by the vector $\mathbf{c}$
If I have done a mistake please correct me.
P.s.: edit to clarify the intent of the question - added future uses - plus fixed set definition.
A cleaner definition of your set $A$ would simply be $$ A = \{ f_1, \dots, f_p \}. $$
Given the functions $f_1, \dots, f_p$, you can certainly define the subspace $C$ spanned by these functions in the (infinite-dimensional) vector space $C(\mathbb R^N)$ of all continuous functions $\mathbb R^N \to \mathbb R$. Using words is probably the best way, but you might also write $$ C = \operatorname{span} \{f_1, \dots, f_p\} \qquad\text{or}\qquad C = \langle f_1, \dots, f_p \rangle. $$
This vector space will be at most $p$-dimensional, but its dimension might be lower if the functions $f_1, \dots, f_p$ are not linear independent. If they’re not independent, different “coordinate” vectors $\mathbf c = [c_1, \dots, c_p]$ will yield the same function (and so these vectors wouldn’t be real coordinates); otherwise, i.e. if the $f_i$ are linearly independent, they form a basis of $C$ and the $\mathbf{c}$ are coordinates wrt this basis.
In any case, $\mathbf c$ will not be an element of $C$: it’s a tuple of real numbers whereas the elements of $C$ are functions. The element of $C$ corresponding to $\mathbf c$ is the function $$ \mathbf x \mapsto f(\mathbf c, \mathbf x). $$
Writing these functions as $f(\mathbf c, \mathbf x)$ is fine, though depending on context, $f_{\mathbf c}(\mathbf x)$ or $f(\mathbf c)(\mathbf x)$ might be nicer.