Difference between revisions of "Index of notation for sets of continuous maps/Index"
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# [[C(X,K)|{{M|C(X,\mathbb{K})}}]] - The ''[[algebra]] of all [[functional|functionals]] on {{M|X}}, where {{M|\mathbb{K} }} is either [[the reals]], {{M|\mathbb{R} }} or [[the complex numbers]], {{M|\mathbb{C} }}, equipped with their usual topology. | # [[C(X,K)|{{M|C(X,\mathbb{K})}}]] - The ''[[algebra]] of all [[functional|functionals]] on {{M|X}}, where {{M|\mathbb{K} }} is either [[the reals]], {{M|\mathbb{R} }} or [[the complex numbers]], {{M|\mathbb{C} }}, equipped with their usual topology. | ||
# [[C(X,F)|{{M|C(X,\mathbb{F})}}]] - '''structure unsure at time of writing''' - set of all ''[[continuous]]'' [[functions]] of the form {{M|f:X\rightarrow\mathbb{F} }} where {{M|\mathbb{F} }} is any [[field]] with an {{link|absolute value|object}}, with the topology that absolute value induces. | # [[C(X,F)|{{M|C(X,\mathbb{F})}}]] - '''structure unsure at time of writing''' - set of all ''[[continuous]]'' [[functions]] of the form {{M|f:X\rightarrow\mathbb{F} }} where {{M|\mathbb{F} }} is any [[field]] with an {{link|absolute value|object}}, with the topology that absolute value induces. | ||
| + | # [[C(K,R)|{{M|C(K,\mathbb{R})}}]] - {{M|K}} must be a ''[[compact]]'' [[topological space]]. Denotes the ''[[algebra]]'' of [[real functionals]] from {{M|K}} to {{M|\mathbb{R} }} - in line with the notation [[C(X,R)|{{M|C(X,\mathbb{R})}}]]. | ||
| + | # [[C(K,C)|{{M|C(K,\mathbb{C})}}]] - {{M|K}} must be a ''[[compact]]'' [[topological space]]. Denotes the ''[[algebra]]'' of [[complex functionals]] from {{M|K}} to {{M|\mathbb{C} }} - in line with the notation [[C(X,C)|{{M|C(X,\mathbb{C})}}]]. | ||
| + | # [[C(K,K)|{{M|C(K,\mathbb{K})}}]] - {{M|K}} must be a ''[[compact]]'' [[topological space]]. Denotes either [[C(K,R)|{{M|C(K,\mathbb{R})}}]] or [[C(K,C)|{{M|C(K,\mathbb{C})}}]] - we do not care/specify the particular field - in line with the notation [[C(X,K)|{{M|C(X,\mathbb{K})}}]]. | ||
| + | # [[C(K,F)|{{M|C(K,\mathbb{F})}}]] - denotes that the space {{M|K}} is a ''[[compact]]'' [[topological space]], the meaning of the field corresponds to the definitions for {{M|C(X,\mathbb{F})}} as given above for that field - in line with the notation [[C(X,F)|{{M|C(X,\mathbb{F})}}]]. | ||
<noinclude> | <noinclude> | ||
==Notes== | ==Notes== | ||
Latest revision as of 06:20, 1 January 2017
Index
- [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
- See also: [ilmath]C(X,\mathbb{K})[/ilmath]
- [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]
- [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].
