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Where is matrix multiplication locally open?
- Source :
- Linear Algebra and its Applications. 517:167-176
- Publication Year :
- 2017
- Publisher :
- Elsevier BV, 2017.
-
Abstract
- Let ( M 1 , d 1 ) , ( M 2 , d 2 ) be metric spaces. A map f : M 1 → M 2 is said to be locally open at an x 1 ∈ M 1 , if for every e > 0 one finds a δ > 0 such that B ( f ( x 1 ) , δ ) ⊂ f ( B ( x 1 , e ) ) ; here B ( x , r ) stands for the closed ball with center x and radius r . We are particularly interested in the following special case: X , Y , Z are normed spaces, the spaces L ( X , Y ) , L ( Y , Z ) , L ( X , Z ) of linear continuous operators are provided with the operator norm, and the map under consideration is the bilinear map ( S , T ) ↦ S ∘ T (from L ( Y , Z ) × ( L ( X , Y ) to L ( X , Z ) ). For which pairs ( S 0 , T 0 ) ∈ L ( Y , Z ) × ( L ( X , Y ) is it locally open? The main result of the paper gives a complete characterization of pairs ( S , T ) at which this map is locally open in the case of finite-dimensional spaces X , Y , Z .
- Subjects :
- Discrete mathematics
Numerical Analysis
Algebra and Number Theory
010102 general mathematics
Center (category theory)
010103 numerical & computational mathematics
Characterization (mathematics)
01 natural sciences
Matrix multiplication
Combinatorics
Metric space
Discrete Mathematics and Combinatorics
Geometry and Topology
0101 mathematics
Bilinear map
Operator norm
Mathematics
Subjects
Details
- ISSN :
- 00243795
- Volume :
- 517
- Database :
- OpenAIRE
- Journal :
- Linear Algebra and its Applications
- Accession number :
- edsair.doi...........13e4e95fba321bc00484ecf813744e25