scholarly journals Error Bound for Conic Inequality in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jiangxing Zhu ◽  
Qinghai He ◽  
Jinchuan Lin
Keyword(s):  
Hilbert Space ◽  
Error Bound ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Local Error ◽  
Local Error Bound ◽  
Local Error Bounds ◽  
Space Case ◽  
Vector Valued Functions ◽  
Vector Valued

We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds.

Spaces of Continuous Vector Functions as Duals

1988 ◽  
Vol 31 (1) ◽  
pp. 70-78 ◽  
Author(s):  
Michael Cambern ◽  
Peter Greim
Keyword(s):  
Hilbert Space ◽  
Weak Topology ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Continuous Vector ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractA well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.

On Learning Vector-Valued Functions

2005 ◽  
Vol 17 (1) ◽  
pp. 177-204 ◽  
Author(s):  
Charles A. Micchelli ◽  
Massimiliano Pontil
Keyword(s):  
Hilbert Space ◽  
Learning Theory ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Support Vector ◽  
Translation Invariant ◽  
Minimal Norm ◽  
Finite Set ◽  
Vector Valued Functions ◽  
Vector Valued

In this letter, we provide a study of learning in a Hilbert space of vector-valued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.

On characterizing the set of martingale measures in discrete time

2015 ◽  
Vol 15 (03) ◽  
pp. 1550017 ◽  
Author(s):  
Abdelkarem Berkaoui
Keyword(s):  
Discrete Time ◽  
Sufficient Conditions ◽  
Sample Space ◽  
Probability Measures ◽  
Finite Sample ◽  
Martingale Measures ◽  
Space Case ◽  
Necessary And Sufficient ◽  
Adapted Process ◽  
Vector Valued

We state necessary and sufficient conditions on a set of probability measures to be the set of martingale measures for a vector valued, bounded and adapted process. In the absence of the maximality condition, we prove the existence of the smallest set of martingale measures. We apply such results to the finite sample space case.

scholarly journals Quadratic reverses of the triangle inequality for Bochner integral in Hilbert spaces

2004 ◽  
Vol 70 (3) ◽  
pp. 451-462
Author(s):  
Sever S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Triangle Inequality ◽  
Bochner Integral ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

Some quadratic reverses of the continuous triangle inequality for the Bochner integral of vector-valued functions in Hilbert spaces are given. Applications of complex-valued functions are provided as well.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

Approximate controllability of stochastic equations in a Hilbert space with fractional Brownian motions

2014 ◽  
Vol 15 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Mo Chen
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Additive Noise ◽  
Sufficient Conditions ◽  
Stochastic Equations ◽  
Hurst Parameter ◽  
Banach Fixed Point Theorem ◽  
Fractional Brownian Motions

In this paper, the approximate controllability for semilinear stochastic equations in Hilbert spaces is studied. The additive noise is the formal derivative of a fractional Brownian motion in a Hilbert space with the Hurst parameter in the interval (½, 1). Sufficient conditions are established. The results are obtained by using the Banach fixed point theorem.

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