Index of notation for sets of continuous maps/Index

Index

1. [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.
2. [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]
3. [ilmath]C(X,\mathbb{R})[/ilmath] - The algebra of all real functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{R} [/ilmath] considered with usual topology
   • See also: [ilmath]C(X,\mathbb{K})[/ilmath]
4. [ilmath]C(X,\mathbb{C})[/ilmath] - The algebra of all complex functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{C} [/ilmath] considered with usual topology
   • See also: [ilmath]C(X,\mathbb{K})[/ilmath]
5. [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.
6. [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.
7. [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].
8. [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].
9. [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].
10. [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].