DF-space Research Articles

The Tensor Product Representation of Polynomials of Weak Type in a DF-Space

LetEandFbe locally convex spaces overCand letP(nE;F)be the space of all continuousn-homogeneous polynomials fromEtoF. We denote by⨂n,s,πEthen-fold symmetric tensor product space ofEendowed with the projective topology. Then, it is well known that each polynomialp∈P(nE;F)is represented as an element in the spaceL(⨂n,s,πE;F)of all continuous linear mappings from⨂n,s,πEtoF. A polynomialp∈P(nE;F)is said to beof weak typeif, for every bounded setBofE,p|Bis weakly continuous onB. We denote byPw(nE;F)the space of alln-homogeneous polynomials of weak type fromEtoF. In this paper, in case thatEis a DF space, we will give the tensor product representation of the spacePw(nE;F).

On topological tensor products of functional Fréchet and DF spaces

A convenient technique for calculating completed topological tensor products of functional Fréchet and DF spaces is developed. The general construction is applied to proving kernel theorems for a wide class of weighted spaces of smooth and entire analytic functions.

Abuilt-in active sensing system-based structural health monitoring technique using statistical pattern recognition

A piezoelectric sensor-based health monitoring technique using a two-step support vector machine (SVM) classifier is developed for railroad track damage identification. Abuilt-in active sensing system composed of two PZT patches was investigated in conjunction with both impedance and guided wave propagation methods to detect two kinds of damage in a railroad track (hole-damage 0.5cm in diameter at the web section and transverse cut damage 7.5cm in length and 0.5cm in depth at the head section). Two damage-sensitive features were separately extracted from each method: a) feature I: root mean square deviations (RMSD) of impedance signatures, and b) feature II: sum of square of wavelet coefficients for maximum energy mode of guided waves. By defining damage indices from these two damagesensitive features, a two-dimensional damage feature (2-D DF) space was made. In order to enhance the damage identification capability of the current active sensing system, a two-step SVM classifier was applied to the 2-D DF space. As a result, optimal separable hyper-planes (OSH) were successfully established by the two-step SVM classifier: Damage detection was accomplished by the first step-SVM, and damage classification was carried out by the second step-SVM. Finally, the applicability of the proposed two-step SVM classifier has been verified by thirty test patterns prepared in advance from the intact state and two damage states.

Q-reflexive Locally Convex Spaces

For a locally convex space E , we use the Aron–Berner extension to define canonical mappings from \widehat\otimes_{s,n,π} E''e into different duals of \mathcal P({}^n E) . We investigate necessary and suffcient conditions for the continuity of these mappings, paying particular attention to three special cases — Fréchet spaces, DF spaces and reflexive A-nuclear spaces. We define Q-reflexive spaces as spaces where a certain canonical mapping can be extended to an isomorphism between \widehat\otimes_{s,n,π} E''e and \overline{(\mathcal P({}^nE), τ_b)'_i} . We find examples of such spaces.

The Space of Countably Simple Bounded Functions with Values in a DF-Space

The Space of Countably Simple Bounded Functions with Values in a DF-Space

Compactness and Weak Compactness in Spaces of Compact-Range Vector Measures

This paper features strong and weak compactness in spaces of vector measures with relatively compact ranges in Banach spaces. Its tools are the measure-operator identification of [16] and [24] and the description of strong and weak compactness in spaces of compact operators in [10], [11], and [29].Given a Banach space X and an algebra of sets, it is shown in [16] that under the usual identification via integration of X-valued bounded additive measures on with X-valued sup norm continuous linear operators on the space of -simple scalar functions, the strongly bounded, countably additive measures correspond exactly to those operators which are continuous for the coarser (locally convex) universal measure topology τ on . It is through the latter identification that the results on strong and weak compactness in [10], [11], and [29] can be applied to X-valued continuous linear operators on the generalized DF space to yield results on strong and weak compactness in spaces of vector measures.

Weakly] compact operators and DF spaces

Weakly] compact operators and DF spaces

On Nonstandard Hulls of Convex Spaces

A nonstandard hull of a TVS (locally convex topological vector space) is a standard TVS constructed from a nonstandard model for [3]. If the nonstandard hulls of a TVS are independent of the non-standard model, we say that the TVS has invariant nonstandard hulls. This is (for complete spaces) the property that every finite element is inflnitesimally close to a standard point. We build on the work of Henson and Moore [4], to show that invariance of nonstandard hulls is a self dual property equivalent to bounded sets being precompact, for F and DF spaces, (see Theorem 4.4).