Index of notation/C

Expression	Status	Meanings	See also
$C(X,Y)$	current	The set of continuous functions between topological spaces. There are many special cases of what $X$ and $Y$ might be, for example: $C(I,X)$ - all paths in $(X,\mathcal{ J })$. 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 inner-product 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: [Expand]Index of notation for sets of continuous maps: $C(X,Y)$ - for topological spaces $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$, $C(X,Y)$ is the set of all continuous maps between them. $C(I,X)$ - $I:\eq[0,1]\subset\mathbb{R}$, set of all paths on a topological space $(X,\mathcal{ J })$ Sometimes written: $C([0,1],X)$ $C(X,\mathbb{R})$ - The algebra of all real functionals on $X$. $\mathbb{R}$ considered with usual topology See also: $C(X,\mathbb{K})$ $C(X,\mathbb{C})$ - The algebra of all complex functionals on $X$. $\mathbb{C}$ considered with usual topology See also: $C(X,\mathbb{K})$ $C(X,\mathbb{K})$ - The algebra of all functionals on $X$, where $\mathbb{K}$ is either the reals, $\mathbb{R}$ or the complex numbers, $\mathbb{C}$, equipped with their usual topology. $C(X,\mathbb{F})$ - structure unsure at time of writing - set of all continuous functions of the form $f:X\rightarrow\mathbb{F}$ where $\mathbb{F}$ is any field with an absolute value, with the topology that absolute value induces. $C(K,\mathbb{R})$ - $K$ must be a compact topological space. Denotes the algebra of real functionals from $K$ to $\mathbb{R}$ - in line with the notation $C(X,\mathbb{R})$. $C(K,\mathbb{C})$ - $K$ must be a compact topological space. Denotes the algebra of complex functionals from $K$ to $\mathbb{C}$ - in line with the notation $C(X,\mathbb{C})$. $C(K,\mathbb{K})$ - $K$ must be a compact topological space. Denotes either $C(K,\mathbb{R})$ or $C(K,\mathbb{C})$ - we do not care/specify the particular field - in line with the notation $C(X,\mathbb{K})$. $C(K,\mathbb{F})$ - denotes that the space $K$ is a compact topological space, the meaning of the field corresponds to the definitions for $C(X,\mathbb{F})$ as given above for that field - in line with the notation $C(X,\mathbb{F})$.