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 Thick sets and quasisymmetric maps

1994 ◽  
Vol 135 ◽  
pp. 121-148 ◽  
Author(s):  
Jussi Väisälä ◽  
Matti Vuorinen ◽  
Hans Wallin
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Product Space ◽  
Finite Sequence ◽  
Closed Ball ◽  
Inner Product Space ◽  
Inner Product ◽  
Real Inner Product Space

1.1. Thickness. Let E be a real inner product space. For a finite sequence of points a0, . . . ,ak in E we let a0. . . ,ak denote the convex hull of the set {a0, . . . , ak}. If these points are affinely independent, the set Δ = a0. . .ak is a k-simplex with vertices a0. . . ,ak. It has a well-defined k-volume written as mk(Δ) or briefly as m(Δ). We are interested in sets A ⊂ E which are “nowhere too flat in dimension k”. More precisely, suppose that A ⊂ E, q > 0 and that k: is a positive integer. We let denote the closed ball with center x and radius r. We say that A is (q, k)-thick if for each x ∈ A and r> 0 such that A\ ≠ there is a k-simplex Δ with vertices in A ∩ such that mk(Δ) ≥ qr.

scholarly journals Spinors and canonical hermitian forms

1988 ◽  
Vol 30 (3) ◽  
pp. 263-270 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Clifford Algebra ◽  
Product Space ◽  
Hermitian Form ◽  
Inner Product Space ◽  
Inner Product ◽  
Spin Representation ◽  
Hermitian Forms ◽  
Real Inner Product Space ◽  
Adjoint Operation

The spaceSof spinors associated to a2m-dimensional real inner product space (V, B) carries a canonical Hermitian form 〈 〉 determined uniquely up to real multiples. This form arises as follows: the complex Clifford algebraC(V) of (V, B) is naturally provided with an antilinear involution; this induces an involution on EndSvia the spin representation; this is the adjoint operation corresponding to 〈 〉.

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 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.

scholarly journals On Orthogonally Additive Functions With Big Graphs

2017 ◽  
Vol 31 (1) ◽  
pp. 57-62
Author(s):  
Karol Baron
Keyword(s):  
Topological Space ◽  
Product Space ◽  
Linear Topological Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Additive Functions ◽  
Big Graphs ◽  
Tychonoff Topology ◽  
Real Inner Product Space ◽  
Orthogonally Additive

Abstract Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

scholarly journals Inner Bogoliubov automorphisms of the minimal C* Weyl algebra

1992 ◽  
Vol 34 (3) ◽  
pp. 263-270
Author(s):  
P. L. Robinson
Keyword(s):  
Clifford Algebra ◽  
Product Space ◽  
Clifford Algebras ◽  
Weyl Algebra ◽  
Inner Product Space ◽  
Inner Product ◽  
Spin Group ◽  
Group Of Units ◽  
Orthogonal Geometry ◽  
Real Inner Product Space

Within the context of orthogonal geometry, isometries of a real inner product space induce Bogoliubov automorphisms of its associated Clifford algebras. The question whether or not such automorphisms are inner is of considerable interest and importance. Inner Bogoliubov automorphisms were fully characterized for the C* Clifford algebra by Shale and Stinespring [14] and for the W* Clifford algebra by Blattner [2]: each case engenders a corresponding notion of spin group, constructed as a group of units inside the Clifford algebra [4].

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