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.

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 A-bilinear forms and generalised A-quadratic forms on unitary left A-modules

1989 ◽  
Vol 39 (1) ◽  
pp. 49-58 ◽  
Author(s):  
C.-S. Lin
Keyword(s):  
Quadratic Form ◽  
Bilinear Form ◽  
Normed Space ◽  
Quadratic Forms ◽  
Product Space ◽  
Inner Product Space ◽  
Vector Spaces ◽  
Inner Product ◽  
Bilinear Forms

In this paper we shall define a generalised A-quadratic form and prove that in some way this form and an A-bilinear form are equivalent to each other. Our result characterises that of Vukman in the sense that we use any n vectors for a fixed n ≥ 2, instead of any two vectors. Consequently, a new generalisation of an inner product space among vector spaces is obtained. This also leads to a new relationship between a 2-inner product space and a 2-normed space.

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.

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.    

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.

scholarly journals Orthogonality in normed spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 449-455 ◽  
Author(s):  
J. R. Partington
Keyword(s):  
Euclidean Space ◽  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Normed Spaces ◽  
Separable Space ◽  
Orthogonality In Normed Spaces

Some properties which different definitions or orthogonality in a normed space can possess are considered. It is shown that orthogonality can be defined on any separable space with many of the properties possessed by the usual orthogonality in an inner-product space, but that the possession of a further property forces the space to be isomorphic to a Euclidean space.

scholarly journals On Carlsson type orthogonality and characterization of inner product spaces

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 859-870 ◽  
Author(s):  
Eder Kikianty ◽  
Sever Dragomir
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Isosceles Orthogonality ◽  
Inner Product Spaces ◽  
Pythagorean Orthogonality

In an inner product space, two vectors are orthogonal if their inner product is zero. In a normed space, numerous notions of orthogonality have been introduced via equivalent propositions to the usual orthogonality, e.g. orthogonal vectors satisfy the Pythagorean law. In 2010, Kikianty and Dragomir [9] introduced the p-HH-norms (1 ? p < ?) on the Cartesian square of a normed space. Some notions of orthogonality have been introduced by utilizing the 2-HH-norm [10]. These notions of orthogonality are closely related to the classical Pythagorean orthogonality and Isosceles orthogonality. In this paper, a Carlsson type orthogonality in terms of the 2-HH-norm is considered, which generalizes the previous definitions. The main properties of this orthogonality are studied and some useful consequences are obtained. These consequences include characterizations of inner product space.

On idempotent affine mappings

1983 ◽  
Vol 93 (3-4) ◽  
pp. 345-348 ◽  
Author(s):  
R. J. H. Dawlings
Keyword(s):  
Vector Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Affine Mapping ◽  
Affine Mappings

SynopsisLet V be a vector space and End (V) the semigroup of endomorphisms of V. An affine mapping of V is a map A: V → V given by xA = xα + a, where a belongs to End (V) and a is some element of V. Let (V) be the semigroup of affine mappings of V.Let E' denote the non-injective idempotents of End (V) and let ℰ denote the idempotents of (V). In this paper 〈ℰ〉 is determined in terms of 〈E′〉 when End (V) consists of all endomorphisms of V and when End (V) only contains the continuous endomorphisms (in which case we restrict V to being an inner product space).

scholarly journals Some Properties of Fuzzy Inner Product Space

2021 ◽  
pp. 2384-2392
Author(s):  
Jehad R. Kider
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Closed Subspace ◽  
Cross Product ◽  
Inner Product Space ◽  
Inner Product ◽  
Orthogonal Complement ◽  
Fuzzy Normed Space ◽  
Direct Sum ◽  
New Type

     Our goal in the present paper is to introduce a new type of fuzzy inner product space. After that, to illustrate this notion, some examples are introduced. Then we prove that that every fuzzy inner product space is a fuzzy normed space. We also prove that the cross product of two fuzzy inner spaces is again a fuzzy inner product space. Next, we prove that the fuzzy inner product is a non decreasing function. Finally, if U is a fuzzy complete fuzzy inner product space and D is a fuzzy closed subspace of U, then we prove that U can be written as a direct sum of D and the fuzzy orthogonal complement    of D.

scholarly journals Boundedness of sign-preserving charges, regularity, and the completeness of inner product spaces

2005 ◽  
Vol 78 (2) ◽  
pp. 199-210 ◽  
Author(s):  
Emmanuel Chetcuti ◽  
Anatolij Dvurečenskij
Keyword(s):  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Completeness Criterion

AbstractWe introduce sign-preserving charges on the system of all orthogonally closed subspaces, F(S), of an inner product space S, and we show that it is always bounded on all the finite-dimensional subspaces whenever dim S = ∞. When S is finite-dimensional this is not true. This fact is used for a new completeness criterion showing that S is complete whenever F(S) admits at least one non-zero sign-preserving regular charge. In particular, every such charge is always completely additive.

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