Inner Products and Norms

2018 ◽  
pp. 129-182 ◽  
Author(s):  
Peter J. Olver ◽  
Chehrzad Shakiban
Keyword(s):  
Inner Products

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

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

scholarly journals All about the ⊥ with its applications in the linear statistical models

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Augustyn Markiewicz ◽  
Simo Puntanen
Keyword(s):  
Statistical Models ◽  
Inner Product ◽  
Real Matrix ◽  
Inner Products ◽  
Column Space ◽  
The Matrix ◽  
Linear Statistical Models ◽  
Extensive List

Abstract For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

Semidefinite Invariant Subspaces: Degenerate Inner Products

2004 ◽  
pp. 467-486
Author(s):  
Christian Mehl ◽  
André C. M. Ran ◽  
Leiba Rodman
Keyword(s):  
Invariant Subspaces ◽  
Inner Products

scholarly journals A Remark on the Homogeneity of Isosceles Orthogonality

2014 ◽  
Vol 2014 ◽  
pp. 1-3 ◽  
Author(s):  
Chan He ◽  
Dan Wang
Keyword(s):  
Isosceles Orthogonality ◽  
Inner Products ◽  
Definition Of ◽  
Homogeneity Of Isosceles Orthogonality

Inspired by the definition of homogeneous direction of isosceles orthogonality, we introduce the notion of almost homogeneous direction of isosceles orthogonality and show that, surprisingly, these two notions coincide. Several known characterizations of inner products are improved.

scholarly journals SOLUTION OF 1D AND 2D POISSON'S EQUATION BY USING WAVELET SCALING FUNCTIONS

2016 ◽  
Vol 15 (2) ◽  
pp. 68
Author(s):  
R. B. Burgos ◽  
H. F. C. Peixoto
Keyword(s):  
Galerkin Method ◽  
Poisson’S Equation ◽  
Scaling Functions ◽  
Poisson's Equation ◽  
Linear System Of Equations ◽  
Inner Products ◽  
Connection Coefficients ◽  
Wavelet Finite Element Method ◽  
Stiffness Matrices ◽  
The Galerkin Method

The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelet scaling functions as a basis in computational analysis holds some promise due to their compact support, orthogonality and localization properties. Daubechies and Deslauriers-Dubuc functions have been successfully used as basis functions in several schemes like the Wavelet- Galerkin Method (WGM) and the Wavelet Finite Element Method (WFEM). Another possible advantage of their use is the fact that the calculation of integrals of inner products of wavelet scaling functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of using approximations by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness matrices and load vectors. In this work, some mathematical foundations regarding wavelet scaling functions, their derivatives and connection coefficients are reviewed. A scheme based on the Galerkin Method is proposed for the direct solution of Poisson's equation (potential problems) in a meshless formulation using interpolating wavelet scaling functions (Interpolets). The applicability of the proposed method and some convergence issues are illustrated by means of a few examples.

Inner Products of Multilinear Vectors

1964 ◽  
Vol 37 (2) ◽  
pp. 93
Author(s):  
H. Randolph Pyle
Keyword(s):  
Inner Products
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