About the Polarity in a Real Linear Space

Affine Independence in Vector Spaces In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set. Finally, we introduce and prove selected properties of the barycentric coordinates.

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On the Hahn-Banach Extension Property

Canadian Mathematical Bulletin ◽

10.4153/cmb-1970-002-4 ◽

1970 ◽

Vol 13 (1) ◽

pp. 9-13

Author(s):

Ting-On To

Keyword(s):

Linear Function ◽

Linear Space ◽

Linear Extension ◽

Normed Linear Space ◽

Extension Property ◽

Bounded Linear ◽

Linear Spaces ◽

Real Linear Space ◽

Real Linear

In this paper, we consider real linear spaces. By (V:‖ ‖) we mean a normed (real) linear space V with norm ‖ ‖. By the statement "V has the (Y, X) norm preserving (Hahn-Banach) extension property" we mean the following: Y is a subspace of the normed linear space X, V is a normed linear space, and any bounded linear function f: Y → V has a linear extension F: X → V such that ‖F‖ = ‖f‖. By the statement "V has the unrestricted norm preserving (Hahn-Banach) extension property" we mean that V has the (Y, X) norm preserving extension property for all Y and X with Y ⊂ X.

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Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range

Canadian Mathematical Bulletin ◽

10.4153/cmb-2003-022-3 ◽

2003 ◽

Vol 46 (2) ◽

pp. 216-228 ◽

Cited By ~ 14

Author(s):

Chi-Kwong Li ◽

Leiba Rodman ◽

Peter Šemrl

Keyword(s):

Linear Space ◽

Numerical Range ◽

Positive Definiteness ◽

Linear Operators ◽

Bounded Linear ◽

Real Linear Space ◽

The Real ◽

Selfadjoint Operators ◽

Linear Maps ◽

Real Linear

AbstractLet H be a complex Hilbert space, and be the real linear space of bounded selfadjoint operators on H. We study linear maps ϕ: → leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps ϕ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption ϕ has the form , for a suitable invertible or unitary T and ξ ∈ {1, −1}, where Xt stands for the transpose of X relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on H. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form αI + K, where α is a scalar and K is compact.

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One-parameter semigroups of subsets of a real linear space

Arkiv för matematik ◽

10.1007/bf02591324 ◽

1960 ◽

Vol 4 (1) ◽

pp. 87-97 ◽

Cited By ~ 13

Author(s):

Hans Rådström

Keyword(s):

Linear Space ◽

Real Linear Space ◽

Real Linear

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ON EXTREME POINTS OF A CERTAIN LINEAR SPACE OF LOCALLY UNIVALENT FUNCTIONS

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.23.1992.4555 ◽

1992 ◽

Vol 23 (4) ◽

pp. 321-325

Author(s):

KUALIDA INAYAT NOOR

Keyword(s):

Linear Space ◽

Unit Disc ◽

Univalent Functions ◽

Extreme Points ◽

Space Structure ◽

Real Linear Space ◽

The Real ◽

Bounded Boundary Rotation ◽

Boundary Rotation ◽

Real Linear

Let $H = (H, \oplus, \odot)$ denote the real linear space of locally univalent normalized functions in the unit disc as defined by Hornich. For $-1\le B <A\le 1$, $k>2$, the classes $V_k[A,B]$ of functions with bounded boundary rotation are introduced and this linear space structure is used to determine the extreme points of the classes $V_k[A,B]$.

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Some Convexity Features Associated with Unitary Orbits

Canadian Journal of Mathematics ◽

10.4153/cjm-2003-004-x ◽

2003 ◽

Vol 55 (1) ◽

pp. 91-111 ◽

Cited By ~ 7

Author(s):

Man-Duen Choi ◽

Chi-Kwong Li ◽

Yiu-Tung Poon

Keyword(s):

Convex Hull ◽

Linear Space ◽

Spectral Properties ◽

Numerical Range ◽

Symmetric Matrices ◽

Hermitian Matrices ◽

Linear Transformations ◽

Unitary Similarity ◽

Similarity Orbit ◽

Real Linear

AbstractLet be the real linear space of n × n complex Hermitian matrices. The unitary (similarity) orbit of C ∈ is the collection of all matrices unitarily similar to C. We characterize those C ∈ such that every matrix in the convex hull of can be written as the average of two matrices in . The result is used to study spectral properties of submatrices of matrices in , the convexity of images of under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

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