Space Of Continuous Linear Functionals Research Articles

$$L_1$$ L 1 -space for a positive operator affiliated with von Neumann algebra

In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an L1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent L1- type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that L1 -type space is isometrically isomorphic to the predual of von Neumann algebra in a natural way. We give a small list of alternate definitions of the seminorm, and a special definition for the case of semifinite von Neumann algebra, in particular. We study order properties of L1-type space, and demonstrate the con- nection between semifinite normal weights and positive elements of this space. At last, we construct a similar L-space for the positive element of C*-algebra, and study the connection between this L-space and the L1 -type space in case when this C*-algebra is a von Neumann algebra.

The dual of Cps(X)

This is a study of the dual space of continuous linear functionals on the function space C ps (X) with a natural norm inherited from a larger Banach space. Here ps denotes the pseudocompact-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X. The lattice structure and completeness of this dual space have been studied. Since this dual space is inherently related to a space of measures, the measure-theoretic characterization of this dual space has been studied extensively. Due to this characterization, a special kind of topological space, called pz-space, has been studied. Finally the separability of this dual space has been studied.

Dyadic distributions

On the basis of the concept of pointwise dyadic derivative dyadic distributions are introduced as continuous linear functionals on the linear space of infinitely differentiable functions compactly supported by the positive half-axis together with all dyadic derivatives. The completeness of the space of dyadic distributions is established. It is shown that a locally integrable function on generates a dyadic distribution. In addition, the space of infinitely dyadically differentiable functions on rapidly decreasing in the neighbourhood of is defined. The space of dyadic distributions of slow growth is introduced as the space of continuous linear functionals on . The completeness of the space is established; it is proved that each integrable function on with polynomial growth at generates a dyadic distribution of slow growth. Bibliography: 25 titles.

Bolzano’s intermediate-value theorem for quasi-holomorphic maps

Abstract We extend Bolzano’s intermediate-value theorem to quasi-holomorphic maps of the space of continuous linear functionals from l p into the scalar field, (0< p<1). This space is isomorphic to l ∞.

On Linear Functionals of Rational Type over H(ID)

AbstractLet H(ID) denote the set of analytic functions in the unit disk ID of the complex plane, endowed with the topology of uniform convergence on compact subsets of ID. Let Λ denote the space of continuous linear functionals over H(ID). In this paper we obtain a characterization of those λ ∈Λ such that where B is a fixed function in H(ID). We also include a discussion of the sharpness of this characterization and an application to some extremal problem concerning Λ.

The Attouch-Wets topology and a characterisation of normable linear spaces

Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.

Orthogonal series representations for generalized functions

A test function space A L wp of smooth weighted L p -functions is constructed whose elements possess orthonormal series expansion converging in the space. Conditions for convergence in A L wp -space of any orthogonal series are obtained. The space of continuous linear functionals on A L wp is investigated and orthogonal series representations for its elements are given. Various special cases of the expansion are discussed, and their applications are pointed out.

Harmonic representations of distributions

This paper is concerned with a characterization of functions which are regular harmonic in the xy plane except on the line y = 0, and have distributional. limits as y -+ 0+ and as y -+ O-. It is shown that such functions satisfy a generalized Green’s formula and may be represented as potentials of singleand double-layer distributions on the line y = 0. This generalizes well-known classical results. Furthermore, it turns out that this class of harmonic functions is identical with the class of functions which are restrictions, to y # 0, of distributions in W(P), whose Laplacians have their support on the x axis. Notation and terminology used here fohow that of Gel’fand and Shilov ES] and Bremermann [2]. Let U be an open subset of R”. The space of test functions on U is the set b(U) of all infinitely differentiable functions with compact support in U. It is given the usual topology [S]. The space of continuous linear functionals on D(U) is denoted by B’(U) and is given the weak topology. Its elements are called (Schwartz) distributions. For the fundamental properties of test functions and distributions, see [2, 5, 10, 121, for example.

Banach Spaces Whose Duals Contain l 1 (Γ) With Applications to the Study of Dual L 1 (μ) Spaces

THEOREM I. If E is a separable Banach space such that E has a complemented subspace isomorphic to 11(r) with r uncountable then E contains a complemented, o(EI, E) closed subspace isomorphic to M(A), the Radon measures on the Cantor set. THEOREM II. If E is a separable Banach space such that E' has a subspace isomorphic to 11(r) with r uncountable, then E contains a subspace isomorphic to 11. THEOREM III. Let E be a Banach space. The following are equivalent: (i) E 'is isomorphic to 1li(r); (ii) every absolutely summing operator on E is nuclear; (iii) every compact, absolutely summing operator on E is nuclear; (iv) if X is a separable subspace of E, then there exists a subspace Y such that X C Y C E and Y 'is isomorphic to 11. THEOREM IV. If E is a 20, space then (i) E is isomorphic to 1l(r) for some set r or (ii) E contains a complemented subspace isomorphic to M(^. COROLLARY. If E is a separable ?P space, then E is (i) finite dimensional, or (ii) isomorphic to 1, or (iii) isomorphic to M(A). COROLLARY. If Ll(u) is isomorphic to the conjugate of a separable Banach space, then LI(C) is isomorphic to 11 or M(A). Introduction. In the past few years, a great deal of information has been obtained about Banach spaces E such that E , the dual of E (the space of continuous linear functionals on E), is isometric to some L1i() = LI(S, 1, I), the Banach space of (equivalence classes of) measurable, absolutely integrable functions on some abstract measure space (S, 1, y). See [91 for the most recent results in this area. There apparently have been very few results characterizing those Banach spaces E such that E' is isomorphic to some LQ(S, i, fL). Closely related to this problem is the problem raised by Dieudonne [21 of determining those L 1(L) spaces which are isomorphic to dual spaces. The study of this problem began with Gelfand who proved that LJ[O, 1 is not isomorphic to a dual space [31; recently, Pe,'czyn'ski proved that if 1i is u-finite and not purely atomic then L(QL) is not isomorphic to a dual space [161. (See [81 and the references of [161 Received by the editors December 27, 1971. AMS (MOS) subject classifications (1969). Primary 4610, 4635.

A note on the inheritance of properties of locally convex spaces by subspaces of countable codimension

A locally convex space E is said to be c-barrelled if every countable weak* bounded subset of its topological dual E' is equicontinuous; to have property (C) if every weak* bounded stubset of E' is relatively weak* compact; to have property (S) if E' is weak* sequentially complete. If a locally convex space possesses any of the above properties, then so do all of its linear subspaces of countable codimension. Examples are furnished to show that the mentioned properties are distinct from each other. In [4], the authors proved that a linear subspace of countable codimension in a barrelled space is barrelled. In this paper, we see that the properties of a space being w-barrelled and having weak*sequentially complete dual are inherited by subspaces of countable codimension. Only a partial result is obtained concerning the seemingly most important property, that of being a Mackey space. The paper concludes with examples to demonstrate that the various concepts discussed are, in fact, disjoint. 1. The notation will be essentially that used by J. Horvath [3]. If (E, F) is a dual pairing (E and F not necessarily separating points), then o-(E, F) will denote the topology on E of pointwise convergence on F. The polar A' of a subset A of E is the set {If F: (a,f)l ? 1 for all aEA }. -r(F, E) will denote the topology on F of uniform convergence on o-(E, F)-compact subsets of E. The vector space of continuous linear functionals on a locally convex space E will be designated E'. A (not necessarily Hausdorff) locally convex space E is said to be barrelled if every closed, balanced, convex, absorbing subset of E is a neighborhood of 0 or, equivalently, if every o-(E', E)-bounded subset of E' is equicontinuous. A locally convex space E is said to be w-barrelled if every countable, o-(E', E)-bounded subset of E' is equicontinuous; to have property (C) if every o(E', E)-bounded subset of E' is relatively o-(E', E)-countably compact; to have property (S) if E' is o-(E', E)-sequentially complete; to be a Mackey space if it has the topology r(E, E'). The codimension of a linear subspace M of Presented to the Society, January 24, 1969; received by the editors June 10, 1970. AMAS 1970 subject classifications. Primary 46A05, 46A15.

Level sets and continuity of conjugate convex functions

A finite-valued convex function on a nonempty convex set C in F can always be extended to a proper convex function on F by assigning it the value + 0o outside of C. Let F and G be real vector spaces in duality with respect to a bilinear functional (x, y) for x e F and y E G (see [1, p. 48]). We shall henceforth assume F and G have each been supplied with a topology compatible with this duality [1, p. 67], so that each can be identified with the space of continuous linear functionals on the other. Unless explicit notice is given, all questions of closure, continuity and boundedness refer to these given topologies. The formulas

Conjugate Banach spaces

The purpose of this note is to present two characterizations of conjugate Banach spaces. More precisely, we present two conditions, each necessary and sufficient for a (real or complex) Banach space to be isomorphic to the conjugate space of a Banach space, and two corresponding conditions for to be equivalent to the conjugate space of a Banach space. Other characterizations, in terms of weak topologies, have been given by Alaoglu ((1), Theorem 2:1, p. 256, and Corollary 2:1, p. 257) and Bourbaki ((4), Chap, IV, §5, exerc. 15c, p. 122). Here, by the conjugate space* of a Banach space we mean ((2), p. 188) the space of continuous linear functionals over . Two Banach spaces and are said to be isomorphic if there is a one-one continuous linear mapping of onto (its inverse is necessarily continuous by the inversion theorem ((2), Théorème 5, p. 41; (6), Theorem 2·13·7, Corollary, p. 29)); they are said to be equivalent if there is a norm-preserving linear mapping of onto .((2), p. 180).