[permalink] [id link]
Every topological vector space X gives a bornology on X by defining a subset to be bounded iff for all open sets containing zero there exists a with.
from
Wikipedia
Some Related Sentences
Every and topological
* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.
* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.
Every group can be trivially made into a topological group by considering it with the discrete topology ; such groups are called discrete groups.
Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.
Every subgroup of a topological group is itself a topological group when given the subspace topology.
Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.
* Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval 1.
Every directed acyclic graph has a topological ordering, an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge.
Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological group.
Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.
Every topological group is an H-space ; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.
Every interior algebra can be represented as a topological field of sets with its interior and closure operators corresponding to those of the topological space.
Every metric space which is ccc is also separable, but in general a ccc topological space need not be separable.
Every locally compact group which is second-countable is metrizable as a topological group ( i. e. can be given a left-invariant metric compatible with the topology ) and complete.
Every and vector
Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.
Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.
Every vector space has a basis, and all bases of a vector space have the same number of elements, called the dimension of the vector space.
Every normed vector space V sits as a dense subspace inside a Banach space ; this Banach space is essentially uniquely defined by V and is called the completion of V.
Every finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).
Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.
Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
Every coalgebra, by ( vector space ) duality, gives rise to an algebra, but not in general the other way.
* Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.
For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".
Many principles that imply the axiom of choice in their general form ( such as " Every vector space has a basis ") become provable in weak subsystems of second-order arithmetic when they are restricted.
Every vector space is free, and the free vector space on a set is a special case of a free module on a set.
Every and space
Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.
* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.
Every node on the Freenet network contributes storage space to hold files, and bandwidth that it uses to route requests from its peers.
* Every Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )
0.360 seconds.