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About 2,633 papers for “Vector-valued function”

Bounded Linear Operators on Banach Function Spaces of Vector-Valued Functions

N. E. Gretsky, J. J. Uhl · Transactions of the American Mathematical Society · 1972

Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.

Cited by 4

On Locally Convex Topological Vector Space Valued Paranormed Function Space Defined by Orlicz Function

NP Pahari · Nepal Journal of Science and Technology · 2014

The aim of this paper is to introduce and study a new class (l∞ (X, Y, Φ, ξ, w , L), HU) of locally convex space Y- valued functions using Orlicz function Φ as a generalization of some of the well known sequence spaces and function spaces. Besides the investigation pertaining to the linear topological structures of the class (l∞ (X, Y, Φ, ξ, w , L), HU) when topologized it with suitable natural paranorm , our primarily interest is to explore the conditions pertaining the containment relation of the class l∞ (X, Y, Φ, ξ, w) in terms of different ξ and w so that such a class of functions is contained in or equal to another class of similar nature. DOI: http://dx.doi.org/10.3126/njst.v14i2.10423 Nepal Journal of Science and Technology Vol. 14, No. 2 (2013) 109-116

Cited by 0Open access

An Intersection Representation for a Class of Anisotropic Vector-valued Function Spaces

Nick Lindemulder · arXiv (Cornell University) · 2019

The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting à la Hedberg$\&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel spaces. In the special case of the classical Lizorkin-Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted $L_{q}$-$L_{p}$-maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin-Triebel spaces occur as spaces of boundary data.

Cited by 0Open access

On the noninferior set for the systems with vector-valued objective function

Kai-ching Chu · IEEE Transactions on Automatic Control · 1970

The necessary and sufficient conditions of the locally noninferior set for the static optimization problem with a vector-valued objective function have been derived. Equality and inequality constraints in the variable space have been discussed to give a more complete formulation. All of these have been done by using the properties of the dual linear inequality systems.

Cited by 37

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Marian Nowak · Proceedings of the American Mathematical Society · 2007

Let 𝐸 be an ideal of 𝐿⁰ over a 𝜎 -finite measure space (Ω,Σ,𝜇) , and let 𝐸^{∼} stand for the order dual of 𝐸 . For a real Banach space (𝑋,|⋅|_{_{𝑋}}) let 𝐸(𝑋) be a subspace of the space 𝐿⁰(𝑋) of 𝜇 -equivalence classes of strongly Σ -measurable functions 𝑓:Ω⟶𝑋 and consisting of all those 𝑓∈𝐿⁰(𝑋) for which the scalar function |𝑓(⋅)|_{_{𝑋}} belongs to 𝐸 . For a real Banach space (𝑌,|⋅|_{_{𝑌}}) a linear operator 𝑇:𝐸(𝑋)⟶𝑌 is said to be order-weakly compact whenever for each 𝑢∈𝐸⁺ the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis StartSet f element-of upper E left-parenthesis upper X right-parenthesis colon double-vertical-bar f left-parenthesis dot right-parenthesis double-vertical-bar Subscript Sub Subscript upper X Subscript Baseline less-than-or-equal-to u EndSet right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi/> <mml:mi>X</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mi>u</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,T(\{f\in E(X):\; \|f(\cdot )\|_{_X}\le u\})\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is relatively weakly compact in 𝑌 . In this paper we examine order-weakly compact operators 𝑇:𝐸(𝑋)⟶𝑌 . We give a characterization of an order-weakly compact operator 𝑇 in terms of the continuity of the conjugate operator of 𝑇 with respect to some weak topologies. It is shown that if (𝐸,|⋅|_{_{𝐸}}) is an order continuous Banach function space, 𝑋 is a Banach space containing no isomorphic copy of 𝑙¹ and 𝑌 is a weakly sequentially complete Banach space, then every continuous linear operator 𝑇:𝐸(𝑋)⟶𝑌 is order-weakly compact. Moreover, it is proved that if (𝐸,|⋅|_{_{𝐸}}) is a Banach function space, then for every Banach space 𝑌 any continuous linear operator 𝑇:𝐸(𝑋)⟶𝑌 is order-weakly compact iff the norm |⋅|_{_{𝐸}} is order continuous and 𝑋 is reflexive. In particular, for every Banach space 𝑌 any continuous linear operator 𝑇:𝐿¹(𝑋)⟶𝑌 is order-weakly compact iff 𝑋 is reflexive.

Cited by 3Open access

Some structural properties of vector valued <i>φ</i>-function sequence space

S N R Gultom, Evalina Herawati · IOP Conference Series Materials Science and Engineering · 2018

The sequence space W(M), where M is an Orlicz function was introduced by Parashar and Choudhary [1] and Maddox [2]. Let f be φ-function and X be a Banach space. In this work, we introduce vector valued sequence space defined by f, denoted by W(X, f). We study some topological properties and inclusion relations of this space.

Cited by 0Open access

Generalization Bounds of Multitask Learning From Perspective of Vector-Valued Function Learning

Chao Zhang, Dacheng Tao, Tao Hu et al. · IEEE Transactions on Neural Networks and Learning Systems · 2020

In this article, we study the generalization performance of multitask learning (MTL) by considering MTL as a learning process of vector-valued functions (VFs). We will answer two theoretical questions, given a small size training sample: 1) under what conditions does MTL perform better than single-task learning (STL)? And 2) under what conditions does MTL guarantee the consistency of all tasks during learning? In contrast to the conventional task-summation based MTL, the introduction of VF form enables us to detect the behavior of each task and the task-group relatedness in MTL. Specifically, the task-group relatedness examines how the success (or failure) of some tasks affects the performance of the other tasks. By deriving the specific deviation and symmetrization inequalities for VFs, we obtain a generalization bound for MTL to the upper bound of the joint probability that there is at least one task with a large generalization gap. To answer the first question, we discuss how the synergic relatedness between task groups affects the generalization performance of MTL and shows that MTL outperforms STL if almost any pair of complementary task groups is predominantly synergic. Moreover, to answer the second question, we present a sufficient condition to guarantee the consistency of each task in MTL, which requires that the function class of each task should not have high complexity. In addition, our findings provide a strategy to examine whether the task settings will enjoy the advantages of MTL.

Cited by 10

Performance evaluation of ethanol fermentor systems using a vector‐valued objective function

S. Tsuji, Kazuyuki Shimizu · Biotechnology and Bioengineering · 1987

It is shown that the performance evaluation using a vector-valued objection function whose components are the product productivity, the product concentration, and the substrate conversion is quite useful in getting deeper insight into the development of new processes and in determining the operating point. Particular attention is focused on the ethanol fermentation using variety of systems such as the conventional chemostat system, multiple fermentor system, cell recycle system, extractive fermentor system, cell recycle system, extractive fermentor system, and immobilized cell system. The contour map and the projection of the noninferior set are used in investigating the performance improvement and the trade-offs among performance indexes.

Cited by 6