scholarly journals *—Inductive limits and partition of unity

1989 ◽  
Vol 12 (3) ◽  
pp. 429-434
Author(s):  
V. Murali
Keyword(s):  
Inductive Limit ◽  
Partition Of Unity ◽  
Vector Spaces ◽  
Inductive Limits ◽  
Topological Vector Spaces ◽  
Limit Space ◽  
Direct Sum

In this note we define and discuss some properties of partition of unity on *-inductive limits of topological vector spaces. We prove that if a partition of unity exists on a *-inductive limit space of a collection of topological vector spaces, then it is isomorphic and homeomorphic to a subspace of a *-direct sum of topological vector spaces.

scholarly journals Linear mappings between topological vector spaces

1972 ◽  
Vol 14 (1) ◽  
pp. 105-118
Author(s):  
B. D. Craven
Keyword(s):  
Inductive Limit ◽  
Vector Spaces ◽  
Continuous Linear ◽  
Additional Structure ◽  
Normed Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

If A and B are locally convex topological vector spaces, and B has certain additional structure, then the space L(A, B) of all continuous linear mappings of A into B is characterized, within isomorphism, as the inductive limit of a family of spaces, whose elements are functions, or measures. The isomorphism is topological if L(A, B) is given a particular topology, defined in terms of the seminorms which define the topologies of A and B. The additional structure on B enables L(A, B) to be constructed, using the duals of the normed spaces obtained by giving A the topology of each of its seminorms separately.

scholarly journals Stability and closed graph theorems in classes of bornological spaces

1982 ◽  
Vol 23 (2) ◽  
pp. 151-162
Author(s):  
T. K. Mukherjee ◽  
W. H. Summers
Keyword(s):  
Inductive Limit ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Closed Graph ◽  
Quotient Spaces ◽  
Direct Sum ◽  
Bornological Space ◽  
Normed Vector ◽  
Locally Convex

In the general theory of locally convex spaces, the idea of inductive limit is pervasive, with quotient spaces and the less obvious notion of direct sum being among the instances. Bornological spaces provide another important example. As is well known (cf. [7]), a Hausdorff locally convex space E is bornological if, and only if, E is an inductive limit of normed vector spaces. Going even further in this direction, a complete Hausdorff bornological space is an inductive limit of Banach spaces.

scholarly journals Bornologiese pseudotopologiese vektorruimtes

1990 ◽  
Vol 9 (1) ◽  
pp. 15-18
Author(s):  
M. A. Muller
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Direct Sums ◽  
Paper Properties ◽  
Topological Vector Spaces ◽  
Quotient Spaces ◽  
Locally Convex

Homological spaces were defined by Hogbe-Nlend in 1971 and pseudo-topological spaces by Fischer in 1959. In this paper properties of bornological pseudo-topological vector spaces are investigated. A characterization of such spaces is obtained and it is shown that quotient spaces and direct sums o f boruological pseudo-topological vector spaces are bornological. Every bornological locally convex pseudo-topological vector space is shown to be the inductive limit in the category of pseudo-topological vector spaces of a family of locally convex topological vector spaces.

Some C*-Algebras with Outer Derivations, II

1974 ◽  
Vol 26 (1) ◽  
pp. 185-189 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Tensor Product ◽  
Finite Number ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Direct Sum ◽  
Finite Dimensional

In this paper we shall consider the class of C*-algebras which are inductive limits of sequences of finite-dimensional C*-algebras. We shall give a complete description of those C*-algebras in this class every derivation of which is inner.Theorem. Let A be a C*-algebra. Suppose that A is the inductive limit of a sequence of finite-dimensional C*-algebras. Then the following statements are equivalent:(i) every derivation of A is inner;(ii) A is the direct sum of a finite number of algebras each of which is either commutative, the tensor product of a finite-dimensional and a commutative with unit, or simple with unit.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

scholarly journals Strictly barrelled disks in inductive limits of quasi-(LB)-spaces

1996 ◽  
Vol 19 (4) ◽  
pp. 727-732
Author(s):  
Carlos Bosch ◽  
Thomas E. Gilsdorf
Keyword(s):  
Convex Space ◽  
Inductive Limit ◽  
Linear Span ◽  
Locally Convex Space ◽  
Minkowski Functional ◽  
Closed Graph ◽  
Inductive Limits ◽  
Locally Convex ◽  
Barrelled Space

A strictly barrelled diskBin a Hausdorff locally convex spaceEis a disk such that the linear span ofBwith the topology of the Minkowski functional ofBis a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.

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