[ilmath]C(X,Y)[/ilmath]

current

The set of continuous functions between topological spaces. There are many special cases of what [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] might be, for example: [ilmath]C(I,X)[/ilmath]  all paths in [ilmath](X,\mathcal{ J })[/ilmath]. These sets often have additional structure (eg, vector space, algebra)
These spaces may not directly be topological spaces, they may be metric spaces, or normed spaces or innerproduct spaces, these of course do have a natural topology associated with them, and it is with respect to that we refer.
  see Index of notation for sets of continuous maps. Transcluded below for convenience:
Index of notation for sets of continuous maps:
 [ilmath]C(X,Y)[/ilmath]  for topological spaces [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath], [ilmath]C(X,Y)[/ilmath] is the set of all continuous maps between them.
 [ilmath]C(I,X)[/ilmath]  [ilmath]I:\eq[0,1]\subset\mathbb{R} [/ilmath], set of all paths on a topological space [ilmath](X,\mathcal{ J })[/ilmath]
 Sometimes written: [ilmath]C([0,1],X)[/ilmath]
 [ilmath]C(X,\mathbb{R})[/ilmath]  The algebra of all real functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{R} [/ilmath] considered with usual topology
 [ilmath]C(X,\mathbb{C})[/ilmath]  The algebra of all complex functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{C} [/ilmath] considered with usual topology
 [ilmath]C(X,\mathbb{K})[/ilmath]  The algebra of all functionals on [ilmath]X[/ilmath], where [ilmath]\mathbb{K} [/ilmath] is either the reals, [ilmath]\mathbb{R} [/ilmath] or the complex numbers, [ilmath]\mathbb{C} [/ilmath], equipped with their usual topology.
 [ilmath]C(X,\mathbb{F})[/ilmath]  structure unsure at time of writing  set of all continuous functions of the form [ilmath]f:X\rightarrow\mathbb{F} [/ilmath] where [ilmath]\mathbb{F} [/ilmath] is any field with an absolute value, with the topology that absolute value induces.
 [ilmath]C(K,\mathbb{R})[/ilmath]  [ilmath]K[/ilmath] must be a compact topological space. Denotes the algebra of real functionals from [ilmath]K[/ilmath] to [ilmath]\mathbb{R} [/ilmath]  in line with the notation [ilmath]C(X,\mathbb{R})[/ilmath].
 [ilmath]C(K,\mathbb{C})[/ilmath]  [ilmath]K[/ilmath] must be a compact topological space. Denotes the algebra of complex functionals from [ilmath]K[/ilmath] to [ilmath]\mathbb{C} [/ilmath]  in line with the notation [ilmath]C(X,\mathbb{C})[/ilmath].
 [ilmath]C(K,\mathbb{K})[/ilmath]  [ilmath]K[/ilmath] must be a compact topological space. Denotes either [ilmath]C(K,\mathbb{R})[/ilmath] or [ilmath]C(K,\mathbb{C})[/ilmath]  we do not care/specify the particular field  in line with the notation [ilmath]C(X,\mathbb{K})[/ilmath].
 [ilmath]C(K,\mathbb{F})[/ilmath]  denotes that the space [ilmath]K[/ilmath] is a compact topological space, the meaning of the field corresponds to the definitions for [ilmath]C(X,\mathbb{F})[/ilmath] as given above for that field  in line with the notation [ilmath]C(X,\mathbb{F})[/ilmath].
