scholarly journals Linearization of holomorphic mappings on fully nuclear spaces with a basis

1994 ◽  
Vol 36 (2) ◽  
pp. 201-208 ◽  
Author(s):  
Seán Dineen ◽  
Pablo Galindo ◽  
Domingo García ◽  
Manuel Maestre
Keyword(s):  
Open Subset ◽  
Convex Space ◽  
Holomorphic Mapping ◽  
Linear Mapping ◽  
Locally Convex Space ◽  
Holomorphic Mappings ◽  
Topological Isomorphism ◽  
Image Position ◽  
Nuclear Spaces ◽  
Locally Convex

In [13] Mazet proved the following result.If U is an open subset of a locally convex space E then there exists a complete locally convex space (U) and a holomorphic mapping δU: U→(U) such that for any complete locally convex space F and any f ɛ ℋ (U;F), the space of holomorphic mappings from U to F, there exists a unique linear mapping Tf: (U)→F such that the following diagram commutes;The space (U) is unique up to a linear topological isomorphism. Previously, similar but less general constructions, have been considered by Ryan [16] and Schottenloher [17].

Some criteria for nuclearity

1986 ◽  
Vol 100 (1) ◽  
pp. 151-159 ◽  
Author(s):  
M. A. Sofi
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequence Spaces ◽  
Bilinear Forms ◽  
Simple Formulation ◽  
Normal Topology ◽  
Nuclear Spaces ◽  
Locally Convex

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

(LF)-spaces with absolute bases

1970 ◽  
Vol 67 (2) ◽  
pp. 283-286 ◽  
Author(s):  
G. Bennett ◽  
J. B. Cooper
Keyword(s):  
Complex Plane ◽  
Real Line ◽  
Convex Space ◽  
Inductive Limit ◽  
Locally Convex Space ◽  
The Real ◽  
Continuous Seminorm ◽  
Image Position ◽  
Locally Convex ◽  
Absolute Basis

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely aswhere ξk ∈K for each k. If this representation converges absolutely, i.e. iffor every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.

The Schwartz property and nuclearity of spaces of smooth and holomorphic functions in infinite dimensions

1990 ◽  
Vol 107 (2) ◽  
pp. 377-385
Author(s):  
Sten Bjon
Keyword(s):  
Uniform Convergence ◽  
Open Subset ◽  
Convex Space ◽  
Holomorphic Functions ◽  
Locally Convex Space ◽  
Infinite Dimensions ◽  
Continuous Convergence ◽  
Convergence Structure ◽  
Locally Convex ◽  
Local Uniform Convergence

In [8] it was shown that a locally convex space E is a Schwartz space if and only if the convergence algebras Hc(U) and He(U) of holomorphic functions on an open subset of E coincide, i.e. if and only if continuous convergence c (see [1]) and the associated equable convergence structure e (= local uniform convergence, see [2, 13]) coincide.

scholarly journals Unitary representations corresponding to measures with bounded support in infinite dimensions

1983 ◽  
Vol 26 (1) ◽  
pp. 67-72
Author(s):  
José E. Galé
Keyword(s):  
Hilbert Space ◽  
Convex Space ◽  
Separable Hilbert Space ◽  
Locally Convex Space ◽  
Unitary Representations ◽  
Unitary Operators ◽  
Infinite Dimensions ◽  
Bilinear Mapping ◽  
Image Position ◽  
Locally Convex

Let E be a real Hausdorff locally convex space with topological dual E′, topologised by the strong topology. Let (x, x′) denote the bilinear mapping defining the duality between E and E′ (x∈E, x′∈E′). By a unitary representation of E′ we mean an operator valued function U(x′) = Ux′. defined on E′, whose values are unitary operators in a separable Hilbert space H such that

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

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.

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

scholarly journals Extending Edgar's ordering to locally convex spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 175-188
Author(s):  
Neill Robertson
Keyword(s):  
Convex Space ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Mackey Topologies

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

scholarly journals A Fenchel-Rockafellar type duality theorem for maximization

1979 ◽  
Vol 20 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Space ◽  
Duality Theorem ◽  
Locally Convex Space ◽  
Bounded Subset ◽  
Convex Functional ◽  
Lower Semi ◽  
Locally Convex

We prove that sup(f-h)(E) = sup(h*-f*)(E*), where f is a proper lower semi-continuous convex functional on a real locally convex space E, h: E → = [-∞, +∞] is an arbitrary-functional and, f*, h* are their convex conjugates respectively. When h = δG, the indicator of a bounded subset G of E, this yields a formula for sup f(G).

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