scholarly journals A Quasi-Hilbert Space and Its Properties

2021 ◽  
Vol 26 (4) ◽  
Author(s):  
Jawad Al-Delfi
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product

This  paper  studies concept  of a  quasi-inner product space  and its  completeness  to get and  prove some properties of quasi-Hilbert spaces. The best examples of  this notion are spaces   where  0<p<∞.

Quasi-inner product spaces of quasi-Sobolev spaces and their completeness

2018 ◽  
pp. 339
Author(s):  
Jawad Kadhim Khalaf Al-Delfi
Keyword(s):  
Hilbert Space ◽  
Sobolev Spaces ◽  
Hilbert Spaces ◽  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces

      Sequences spaces  , m  ,  p  have called quasi-Sobolev spaces were  introduced   by Jawad . K. Al-Delfi in 2013  [1]. In this  paper , we deal with notion of  quasi-inner product  space  by using concept of  quasi-normed  space which is generalized  to normed space and given a  relationship  between  pre-Hilbert space and a  quasi-inner product space with important  results   and   examples.  Completeness properties in quasi-inner   product space gives  us  concept of  quasi-Hilbert space .  We show  that ,  not  all  quasi-Sobolev spaces  ,  are  quasi-Hilbert spaces. The  best  examples which are  quasi-Hilbert spaces and Hilbert spaces  are , where  m  . Finally, propositions, theorems and examples are our own unless otherwise referred.    

scholarly journals Sub-Exact Sequence on Hilbert Space

2020 ◽  
Vol 55 (6) ◽  
Author(s):  
Bernadhita H. S. Utami ◽  
Fitriani ◽  
Mustofa Usman ◽  
Warsono ◽  
Jamal Ibrahim Daoud
Keyword(s):  
Hilbert Space ◽  
Exact Sequence ◽  
Vector Space ◽  
Direct Summand ◽  
Cauchy Sequence ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Isometric Isomorphism

The notion of the sub-exact sequence is the generalization of exact sequence in algebra, particularly on a module. A module over a ring R is a generalization of the notion of vector space over a field F. A Hilbert space refers to a special vector space over a field F when we have a complete inner product space. The space is complete if every Cauchy sequence converges. Now, we introduce the sub-exact sequence on a Hilbert space, which can be useful later in statistics. This paper is aimed at investigating the properties of the sub-exact sequence and their ratio to direct summand on a Hilbert space. As the result, we obtain two properties of isometric isomorphism sub-exact sequence on a Hilbert space.

scholarly journals SISTEM ORTONORMAL DALAM RUANG HILBERT

2014 ◽  
Vol 8 (2) ◽  
pp. 19-26
Author(s):  
Zeth A. Leleury
Keyword(s):  
Hilbert Space ◽  
Quadratic Form ◽  
Vector Space ◽  
Integral Equations ◽  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Finite Dimensional

Hilbert space is one of the important inventions in mathematics. Historically, the theory of Hilbert space originated from David Hilbert’s work on quadratic form in infinitely many variables with their applications to integral equations. This paper contains some definitions such as vector space, normed space and inner product space (also called pre-Hilbert space), and which is important to construct the Hilbert space. The fundamental ideas and results are discussed with special attention given to finite dimensional pre-Hilbert space and some basic propositions of orthonormal systems in Hilbert space. This research found that each finite dimensional pre- Hilbert space is a Hilbert space. We have provided that a finite orthonormal systems in a Hilbert space X is complete if and only if this orthonormal systems is a basis of X.

Completeness of P-Summable as Hilbert Space with Its Dual Space

2021 ◽  
pp. 51-56
Author(s):  
Muhammad Ryan Sanusi ◽  
Endang Rusyaman ◽  
Diah Chaerani
Keyword(s):  
Hilbert Space ◽  
Vector Space ◽  
Cauchy Sequence ◽  
Product Space ◽  
Dual Space ◽  
Inner Product Space ◽  
Vector Spaces ◽  
Inner Product ◽  
Space Data

Hilbert space is a complete inner product space, meaning that each Cauchy sequence converges to a point in that space. One of the vector spaces that will be examined as the inner product space is p-summable space. The inner product space is a subset of vector spaces that have special properties that must be fulfilled. One way to prove vector space is the inner product space is to use parallelogram equality theorems. After it is known that the vector space is the inner product space, the completeness of the space will be proven using the dual space. The space used is the p-summable space, data that can be changed in a sequence form will be usable in this study. The results of this study will be useful as another application in determining a Hilbert space by using a method that is different from the definition. The analysis used will show comparison of the speed of completion accuracy will be a benchmark in this study, so that will be a new reference in determining a space is Hilbert space.

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

Sign in / Sign up

Export Citation Format

Share Document