DuaL Spaces, Norms, and Inner Products

1998 ◽  
pp. 73-88
Author(s):  
David F. Delchamps
Keyword(s):  
Dual Spaces ◽  
Inner Products

scholarly journals Priestley duality for MV-algebras and beyond

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wesley Fussner ◽  
Mai Gehrke ◽  
Samuel J. van Gool ◽  
Vincenzo Marra
Keyword(s):  
Large Class ◽  
Order Conditions ◽  
Enriched Environment ◽  
Priestley Duality ◽  
Distributive Lattices ◽  
First Order ◽  
Dual Spaces ◽  
Binary Operations ◽  
New Perspective ◽  
Mv Algebras

Abstract We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

scholarly journals Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Angela A. Albanese ◽  
Claudio Mele
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Strong Operator ◽  
Dual Spaces ◽  
Structure Theorems ◽  
The Fourier Transform ◽  
Decreasing Functions

AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.

scholarly journals When are maps preserving semi-inner products linear?

2021 ◽  
Author(s):  
Paweł Wójcik
Keyword(s):  
Hilbert Space ◽  
Infinite Dimensions ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Inner Products ◽  
Finite Dimensional ◽  
Uniformly Smooth ◽  
Non Linear ◽  
Extra Assumption ◽  
Continuous Injection

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

Support Vector Machines V: Exploiting Inner Products

NIR news ◽  
2005 ◽  
Vol 16 (3) ◽  
pp. 4-6 ◽  
Author(s):  
Tom Fearn ◽  
Donald J. Dahm
Keyword(s):  
Support Vector Machines ◽  
Support Vector ◽  
Inner Products ◽  
Vector Machines

Conformally self-dual spaces and Maxwell's equations

1984 ◽  
Vol 106 (3) ◽  
pp. 125-129 ◽  
Author(s):  
C.P. Boyer ◽  
J.F. Plebański
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Dual Spaces

Dual spaces ofJB *-triples and the Radon-Nikodym property

1991 ◽  
Vol 208 (1) ◽  
pp. 327-334 ◽  
Author(s):  
L. J. Bunce ◽  
C. -H. Chu
Keyword(s):  
Dual Spaces ◽  
Nikodym Property

Numerical Evaluation of Reproducing Kernel Hilbert Space Inner Products

2009 ◽  
Vol 57 (3) ◽  
pp. 1227-1233 ◽  
Author(s):  
A. Oya ◽  
J. Navarro-Moreno ◽  
J.C. Ruiz-Molina
Keyword(s):  
Hilbert Space ◽  
Numerical Evaluation ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Inner Products
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