Idempotent Functional Analysis: An Algebraic Approach

By G. Litvinov, V. Maslov, G. Shpiz. 2000

237 citations recorded by the source corpus.

This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d6898bc2e86e9fb91a37cc9de2b971ab46a9b12

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Ricardo D. Katz
2. Stephane Gaubert
3. Cormac Walsh
4. Marianne Akian

In Mathematical Physics

1. Stephane Gaubert
2. Ricardo D. Katz
3. Cormac Walsh
4. Marianne Akian
5. Jeremy Gunawardena

Citations in the corpus

• Best approximation in max-plus semimodules☆ (2010, background): “by [8], Theorem 36; see also [17]), we can also write”
• Inclines and incline matrices: a survey (2004, background): “The papers [26,27] give a view of mathematics over idempotent semirings as a dequantization or classical limit of traditional mathematics as something like a Planck constant goes to zero.”
• Minimal half-spaces and external representation of tropical polyhedra (2009, methodology): “It was studied by Litvinov, Maslov, and Shpiz [29], in relation to problems of calculus of variations, and by Cohen, Gaubert, and Quadrat [12, 13], motivated by discrete event system problems (max-plus polyhedra represent invariant spaces of max-plus linear dynamical systems [11]).”
• The max-plus Martin boundary (2004, background): “There exists a correspondence between classical and idempotent analysis, which was brought to light by Maslov and his collaborators [Mas87, MS92, KM97, LMS01 ].”
• Linear independence over tropical semirings and beyond (2008, background): “Max-plus structures have appeared in relation with in a variety of fields, like operations research and optimization [Vor67, CG79, GM08b], discrete event systems [BCOQ92, CGQ99], automata theory (see [Pin98] and its references), quasi-classic asymptotics [Mas87], Hamilton-Jacobi partial differential”