totally bounded वाक्य
उदाहरण वाक्य
मोबाइल
- It remains true ( that is, the proof does not require choice ) that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded.
- It remains true ( that is, the proof does not require choice ) that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded.
- Then it becomes a theorem that a space is totally bounded if and only if it is precompact . ( Separating the definitions in this way is useful in the absence of the axiom of choice; see the next section .)
- But it is no longer true ( that is, the proof requires choice ) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.
- But it is no longer true ( that is, the proof requires choice ) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.
- And in the special case of an arbitrary compact metric space " X " every bounded subspace of an arbitrary metric space " Y " aimed at " X " is totally bounded ( i . e . its metric completion is compact ).
- X ^ \ star is defined as the space of all linear continuous functionals f : X \ to \ mathbb { C } endowed with the topology of uniform convergence on totally bounded sets in " X ", and the " second dual space"
- There is a complementary relationship between total boundedness and the process of Cauchy completion : A uniform space is totally bounded if and only if its Cauchy completion is totally bounded . ( This corresponds to the fact that, in Euclidean spaces, a set is bounded if and only if its closure is bounded .)
- There is a complementary relationship between total boundedness and the process of Cauchy completion : A uniform space is totally bounded if and only if its Cauchy completion is totally bounded . ( This corresponds to the fact that, in Euclidean spaces, a set is bounded if and only if its closure is bounded .)
- For any two stereotype spaces X and Y the " stereotype space of operators " \ text { Hom } ( X, Y ) from X into Y, is defined as the pseudosaturation of the space \ text { L } ( X, Y ) of all linear continuous maps \ varphi : X \ to Y endowed with the topology of uniform convergeance on totally bounded sets.
- Similarly, one can replace the class of bounded ( and totally bounded ) subsets in X in the definition of dual space X, by other classes of subsets, for example, by the class of compact subsets in X-- the spaces defined by the corresponding reflexivity condition are called " reflective ", and they form an even wider class than "'Ste "', but it is not clear ( 2012 ), whether this class forms a category with properties similar to those of "'Ste " '.
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