Difference between revisions of "The set of continuous functions between topological spaces"
From Maths
(Created page with "{{Stub page|grade=A*|msg=Proper stub, created just to save a link from being red}} ==Definition== Let {{Top.|X|J}} and {{Top.|Y|K}} be topological spaces. Then {{M|C(X,Y)}...") |
m (→See also: Adding index link) |
||
| Line 12: | Line 12: | ||
* [[The space of all continuous linear maps]] - denoted {{M|\mathcal{L}(V,W)}} for {{plural|vector space|s}} {{M|V}} and {{M|W}} usually over either the [[field]] of [[the reals]] or [[field of complex numbers|complex numbers]]<ref group="Note">Both {{M|V}} and {{M|W}} must be over the same field</ref> | * [[The space of all continuous linear maps]] - denoted {{M|\mathcal{L}(V,W)}} for {{plural|vector space|s}} {{M|V}} and {{M|W}} usually over either the [[field]] of [[the reals]] or [[field of complex numbers|complex numbers]]<ref group="Note">Both {{M|V}} and {{M|W}} must be over the same field</ref> | ||
** [[The space of all linear maps]] - denote {{M|L(V,W)}}, for {{M|V}} and {{M|W}} vector spaces over the same field | ** [[The space of all linear maps]] - denote {{M|L(V,W)}}, for {{M|V}} and {{M|W}} vector spaces over the same field | ||
| + | * [[Index of spaces, sets and classes]] | ||
| + | |||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Latest revision as of 05:02, 3 November 2016
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Proper stub, created just to save a link from being red
Contents
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces. Then [ilmath]C(X,Y)[/ilmath] denotes the set of all continuous functions from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath], with respect to the topologies: [ilmath]\mathcal{J} [/ilmath] and [ilmath]\mathcal{K} [/ilmath].
That is to say:
- [ilmath]\big(f\in C(X,Y)\big)\iff\big(f:X\rightarrow Y\text{ is a continuous function}\big)[/ilmath]
See also
- Subsets:
- [ilmath]C([0,1],X)[/ilmath] - all paths in [ilmath]X[/ilmath]
- [ilmath]\Omega(X,b)[/ilmath] - all loops in [ilmath]X[/ilmath] based at [ilmath]b\in X[/ilmath]
- The space of all continuous linear maps - denoted [ilmath]\mathcal{L}(V,W)[/ilmath] for vector spaces [ilmath]V[/ilmath] and [ilmath]W[/ilmath] usually over either the field of the reals or complex numbers[Note 1]
- The space of all linear maps - denote [ilmath]L(V,W)[/ilmath], for [ilmath]V[/ilmath] and [ilmath]W[/ilmath] vector spaces over the same field
- Index of spaces, sets and classes
Notes
- ↑ Both [ilmath]V[/ilmath] and [ilmath]W[/ilmath] must be over the same field
References
Grade: A
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
The message provided is:
Find some!
