Vector Spaces

2021 ◽  
pp. 47-102
Author(s):  
Adel N. Boules
Keyword(s):  
Invariant Subspaces ◽  
Extensive Study ◽  
Vector Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Direct Sums ◽  
Infinite Dimensional ◽  
Quotient Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional

The first three sections of this chapter provide a thorough presentation of the concepts of basis and dimension. The approach is unified in the sense that it does not treat finite and infinite-dimensional spaces separately. Important concepts such as algebraic complements, quotient spaces, direct sums, projections, linear functionals, and invariant subspaces make their first debut in section 3.4. Section 3.5 is a brief summary of matrix representations and diagonalization. Then the chapter introduces normed linear spaces followed by an extensive study of inner product spaces. The presentation of inner product spaces in this section and in section 4.10 is not limited to finite-dimensional spaces but rather to the properties of inner products that do not require completeness. The chapter concludes with the finite-dimensional spectral theory.

scholarly journals A generalization of Lucas' theorem to vector spaces

1993 ◽  
Vol 16 (2) ◽  
pp. 267-276 ◽  
Author(s):  
Neyamat Zaheer
Keyword(s):  
Critical Points ◽  
Vector Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Linear Combinations ◽  
Inner Product Spaces ◽  
Complex Valued ◽  
Vector Valued

The classical Lucas' theorem on critical points of complex-valued polynomials has been generalized (cf. [1]) to vector-valued polynomials defined onK-inner product spaces. In the present paper, we obtain a generalization of Lucas' theorem to vector-valued abstract polynomials defined on vector spaces, in general, which includes the above result of the author [1] inK-inner product spaces. Our main theorem also deduces a well-known result due to Marden on linear combinations of polynomial and its derivative. At the end, we discuss some examples in support of certain claims.

scholarly journals Moore-Penrose inverse of some linear maps on infinite-dimensional vector spaces

2020 ◽  
Vol 36 (36) ◽  
pp. 570-586 ◽  
Author(s):  
Fernando Pablos Romo ◽  
Víctor Cabezas Sánchez
Keyword(s):  
Linear System ◽  
Vector Spaces ◽  
Inner Product ◽  
Dimensional Vector ◽  
Product Spaces ◽  
Infinite Dimensional ◽  
Inner Product Spaces ◽  
Linear Maps ◽  
Infinite Linear System ◽  
Infinite Linear

The aim of this work is to characterize linear maps of infinite-dimensional inner product spaces where the Moore-Penrose inverse exists. This MP inverse generalizes the well-known Moore-Penrose inverse of a matrix $A\in \text{Mat}_{n\times m} ({\mathbb C})$. Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, the least norm solution of an infinite linear system from the Moore-Penrose inverse offered is studied.

scholarly journals Functional Equations Related to Inner Product Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Choonkil Park ◽  
Won-Gil Park ◽  
Abbas Najati
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
Vector Spaces ◽  
Inner Product ◽  
Ulam Stability ◽  
Product Spaces ◽  
Real Vector ◽  
Inner Product Spaces

LetV,Wbe real vector spaces. It is shown that an odd mappingf:V→Wsatisfies∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi)for allx1,…,x2n∈Vif and only if the odd mappingf:V→Wis Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

scholarly journals The Construction of Hilbert Spaces over the Non-Newtonian Field

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Hilbert Spaces ◽  
Euclidean Geometry ◽  
Vector Spaces ◽  
Inner Product ◽  
Complex Structures ◽  
The Novel ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Classical Calculus ◽  
Cartesian Plane

Although there are many excellent ways to present the principle of the classical calculus, the novel presentations probably lead most naturally to the development of the non-Newtonian calculi. In this paper we introduce vector spaces over real and complex non-Newtonian field with respect to the *-calculus which is a branch of non-Newtonian calculus. Also we give the definitions of real and complex inner product spaces and study Hilbert spaces which are special type of normed space and complete inner product spaces in the sense of *-calculus. Furthermore, as an example of Hilbert spaces, first we introduce the non-Cartesian plane which is a nonlinear model for plane Euclidean geometry. Secondly, we give Euclidean, unitary, and sequence spaces via corresponding norms which are induced by an inner product. Finally, by using the *-norm properties of complex structures, we examine Cauchy-Schwarz and triangle inequalities.

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.

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