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})}}]]. | ||
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==Notes== | ==Notes== | ||
Latest revision as of 06:20, 1 January 2017
Index
- 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}).
