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A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set S but also one of the set S as subset of P. A bounded poset P (that is, by itself, not as subset) is one that has a least element and a greatest ...
The collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of .. A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some . [1] The set of all bounded subsets of trivially forms a fundamental system of bounded sets of .
A subset S of a uniform space X is totally bounded if and only if, for any entourage E, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, E replaces the "size" ε, and a subset is of size E if its Cartesian square is a subset of E.) [4]
A family of subsets of a topological vector space is said to be uniformly bounded in , if there exists some bounded subset of such that , which happens if and only if is a bounded subset of ; if is a normed space then this happens if and only if there exists some real such that ‖ ‖.
A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it. In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded. Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of ...
More generally, by the bounded inverse theorem, T is not invertible if it is not bounded below; that is, if there is no c > 0 such that ||Tx|| ≥ c||x|| for all x ∈ X. So the spectrum includes the set of approximate eigenvalues , which are those λ such that T - λI is not bounded below; equivalently, it is the set of λ for which there is a ...
Pavel Urysohn. In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma [1]) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
Under this topology, a net in () converges to () if and only if for every multi-index with | | < + and every compact , the net of partial derivatives () converges uniformly to on . [3] For any {,,, …,}, any (von Neumann) bounded subset of + is a relatively compact subset of (). [4] In particular, a subset of () is bounded if and only if it is ...