separable space वाक्य
उदाहरण वाक्य
मोबाइल
- A set of density 5 ! 0 is a separable space.
- The product of at most continuum many separable spaces is a separable space.
- The product of at most continuum many separable spaces is a separable space.
- For, is also a separable space.
- A space whose set has a countable, dense subset is called a separable space.
- From this perspective, the natural state space of a boson might seem to be a non-separable space.
- In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.
- In practical terms, this means that theorems of algebra and combinatorics are restricted to countable structures, while theorems of analysis and topology are restricted to separable spaces.
- Non-separable Banach spaces cannot embed isometrically in the separable space, but for every Banach space, one can find a Banach & ndash; Alaoglu theorem.
- If " X " is a compact separable space, then the space of finite signed Baire measures is the dual of the real Banach space of all continuous real-valued functions on " X ", by the Riesz Markov Kakutani representation theorem.
- Though there is a theory for von Neumann algebras on non-separable Hilbert spaces ( and indeed much of the general theory still holds in that case ) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces.
- If " H " 1 is a separable space ( in particular, if it is a Euclidean space ) the result is true in Zermelo Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient.
- Marczewski proved that the topological dimension, for arbitrary metrisable separable space " X ", coincides with the Hausdorff dimension under one of the metrics in " X " which induce the given topology of " X " ( while otherwise the Hausdorff dimension is always greater or equal to the topological dimension ).
- If the requirement for the countable chain condition is replaced with the requirement that " R " contains a countable dense subset ( i . e ., " R " is a separable space ) then the answer is indeed yes : any such set " R " is necessarily order-isomorphic to "'R "'( proved by Cantor ).
- :: : I am going to stop now in case I get too carried away; ) but these problems illustrate the diversity of general topology and how simple ideas such as the OP's question may lead to such interesting questions ( for instance, in question one, we know that every separable space satisfies the countable chain condition; the obvious question is whether there exists a non-separable space which satisfies the countable chain condition ( that is, does the converse hold ? ).
- :: : I am going to stop now in case I get too carried away; ) but these problems illustrate the diversity of general topology and how simple ideas such as the OP's question may lead to such interesting questions ( for instance, in question one, we know that every separable space satisfies the countable chain condition; the obvious question is whether there exists a non-separable space which satisfies the countable chain condition ( that is, does the converse hold ? ).
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