Search results
Results From The WOW.Com Content Network
[0, 1] 2 is a totally bounded space because for every ε > 0, the unit square can be covered by finitely many open discs of radius ε. A metric space (,) is totally bounded if and only if for every real number >, there exists a finite collection of open balls of radius whose centers lie in M and whose union contains M.
Note that these measures of non-compactness are useless for subsets of Euclidean space R n: by the Heine–Borel theorem, every bounded closed set is compact there, which means that γ(X) = 0 or ∞ according to whether X is bounded or not. Measures of non-compactness are however useful in the study of infinite-dimensional Banach spaces, for ...
The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space is bounded. To see this, start with a finite cover by r-balls for some arbitrary r. Since the subset of M consisting of the centers of these balls is finite, it has finite diameter, say D.
In other words, the inclusion in the bidual () = is isomorphic to the inclusion of the space of countably additive μ-a.c. bounded measures inside the space of all finitely additive μ-a.c. bounded measures.
The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
In other words, for every there exists such that (,). If furthermore C is a subset of K , then it is an r-internal covering . The external covering number of K , denoted N r ext ( K ) {\displaystyle N_{r}^{\text{ext}}(K)} , is the minimum cardinality of any external covering of K .
is a function space.Its elements are the essentially bounded measurable functions. [2]More precisely, is defined based on an underlying measure space, (,,). Start with the set of all measurable functions from to which are essentially bounded, that is, bounded except on a set of measure zero.