Convergence of Measures and Metrizability: Difference between revisions
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(Created page with "This article should address metrizability for both narrow and wide convergence. ==General Functional Analysis Refs== *Ambrosio, Gigli, Savaré (107-108), Brezis (72-76) ==Na...") |
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==Narrow Convergence== | ==Narrow Convergence== | ||
*Ambrosio, Gigli, Savaré ( | *Unit ball of dual space of <math>C_b(X)</math> is, in general, not metrizable: [[https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007%2F978-1-4757-3828-5 Conway, ''A Course in Functional Analysis'', Theorem 6.6]] | ||
*[[https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007%2F978-1-4757-3828-5 Conway, ''A Course in Functional Analysis'', | *It is metrizable when restricted to probability measures: Ambrosio, Gigli, Savaré (106-108) | ||
==Wide Convergence== | |||
*On the other hand, if X is a metrizable locally compact space that is σ-compact, then <math>C_0(X)</math> is separable, [[https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007%2F978-1-4757-3828-5 Conway, ''A Course in Functional Analysis'', III. Banach Spaces, exercise 14]] | |||
Revision as of 19:50, 13 April 2020
This article should address metrizability for both narrow and wide convergence.
General Functional Analysis Refs
- Ambrosio, Gigli, Savaré (107-108), Brezis (72-76)
Narrow Convergence
- Unit ball of dual space of is, in general, not metrizable: [Conway, A Course in Functional Analysis, Theorem 6.6]
- It is metrizable when restricted to probability measures: Ambrosio, Gigli, Savaré (106-108)
Wide Convergence
- On the other hand, if X is a metrizable locally compact space that is σ-compact, then is separable, [Conway, A Course in Functional Analysis, III. Banach Spaces, exercise 14]
