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 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.

Self-adjoint operators on inner product spaces over fields of power series

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Hans Keller ◽  
Ochsenius Herminia
Keyword(s):  
Power Series ◽  
Adjoint Operator ◽  
Inner Product ◽  
Closed Field ◽  
Product Spaces ◽  
Binary Forms ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Orthogonal Decompositions ◽  
Adjoint Operators

AbstractTheorems on orthogonal decompositions are a cornerstone in the classical theory of real (or complex) matrices and operators on ℝn. In the paper we consider finite dimensional inner product spaces (E, ϕ) over a field K = F((χ 1, ..., x m)) of generalized power series in m variables and with coefficients in a real closed field F. It turns out that for most of these spaces (E, ϕ) every self-adjoint operator gives rise to an orthogonal decomposition of E into invariant subspaces, but there are some salient exceptions. Our main theorem states that every self-adjoint operator T: (E, ϕ) → (E, ϕ) is decomposable except when dim E is a power of 2 with exponent at most m, and ϕ is a tensor product of pairwise inequivalent binary forms. In the exceptional cases we provide an explicit description of indecomposable operators.

scholarly journals Norms on Quotient Spaces of The 2-Inner Product Space

2021 ◽  
Vol 3 (2) ◽  
pp. 80
Author(s):  
Harmanus Batkunde
Keyword(s):  
Product Space ◽  
Independent Set ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Quotient Spaces ◽  
Inner Product Spaces ◽  
Inner Products

This paper discussed about construction of some quotients spaces of the 2-inner product spaces. On those quotient spaces, we defined an inner product with respect to a linear independent set. These inner products was derived from the -inner product. We then defined a norm which induced by the inner product in these quotient spaces.

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.

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