MATHEMATICS I

IN ECONOMICS

1. Introduction and Examples.
2. Vector Spaces.
    2.1. Definitions and examples ([12]: 2.2.)
    2.2. Subspaces ([12]: 2.3.)
    2.3 Conexity and cones ([12]: 2.4.)
    2.4 Linear independence and dimension ([12]: 2.5.)
3. Normed Spaces
    3.1 Definitions and examples ([12]: 2.6.)
    3.2 Tpology in normed spaces ([12]: 2.7, 2.8, 2.9.)
    3.3. The spaces l^p and L^p ([12]: 2.10.)
    3.4. Banach spaces ([12]: 2.11, 2.12.)
    3.5 Maximization and compactness ([12]: 2.13)
4. Hilbert spaces
    4.1. Inner products ([12]: 3.2.)
    4.2. The projection Theorem ([12]: 3.3.)
    4.3. Orthogonal complements ([12]: 3.4.)
    4.4. The Gram-Schmidt procedure ([12]: 3.5.)
    4.5. The dual problem ([12]: 3.10.)
    4.6 Minimal distance to a convex set ([12]: 3.12.)
5. Dual Spaces
    5.1. Basic concepts ([12]: 5.2.)
    5.2. Examples ([12]: 5.3.)
    5.3. Extensions of linear functionals. The Hahn-Banach Theorem ([12]: 5.4.)
    5.4. The dual of C ([a,b] ([12]: 5.5.)
    5.5. Orthogonal complements ([12]: 5.7.)
    5.6. Minimun norms problems ([12]: 5.8.)
    5.7. Applications and examples ([12]: 5.9.)
    5.8. Weak convergence ([12]: 5.10.)
    5.9. Geometric form of the Hahn-Banach Theorem ([12]: 5.11, 5.12, 5.13.)
6 Linear Operators
    6.1. Fundamental properties ([12]: 6.2.)
    6.2. Inverses of linear operators ([12]: 6.3, 6.4.)
    6.3. Adjoints ([12]: 6.5, 6.6, 6.8.)
    6.4. Optimizacion in Hilbert spaces ([12]: 6.9, 6.10)
7. Optimization in Normed Spaces
    7.1 Introduction
    7.2 Differentiation in Normed Spaces ([12]: 7.2, 7.3.)
    7.3. The implicit and Inverse Theorems in Normed Spaces ([12]: 9.2.)
    7.4. Extrema. First order conditions for non-constrained optimization ([12]: 7.4.)
    7.5. Optimization with equality constrains. Lagrange Multiplier ([12]: 9.3.)
    7.6. Optimizacion with inequality constrains. Kuhn-Tucker Theorem. ([12]: 9.4.)
8. Calculus of Variations
    8.1 Introduction
    8.2. Problems with fixed end points. Euler-Lagrange equations. Second order conditions. legendre condition ([12]: 7.5, ([1] 2.3.)
    8.3. Problems with variable end points ([12]: 7.6, ([1] 2.4. chapter 3)
9. Optimal Control Theory in continuous time.
    9.1. Introduction ([1]: 4.1, 4.2.)
    9.2. Basic necessary conditions ([12]: 9.5)
    9.3. Pontryagin's Maximum Principle ([12]: 9.6), ([1]: 4.3).
    9.4. Economic interpretation of Pontryagin's Maximum Principle
    9.5. Sufficient conditions ([1]: 4.6.)
 

• Professor: F. Marhuenda. Office: 15.2.43. Tel.: 91 624 93 66. E-mail: marhuend@eco.uc3m.es
• Office hours. Tuesdays: 15:00h to 17:00h and Thursdays 11:00 to 13:00
• Course evaluation will be based on regular homework assignment and a Final Exam.
• Academic honesty: The rules of the university about academic honesty will be fully enforced. You may use any references for help. You may also discuss (and you are encouraged to do so) the problems with other classmates. But you should give proper credit to your sources (like: "the method followed is based on example A in book X" and I have discussed problem number A"). Failing to do so will be considered cheating. Also, you should understand the problem enough so you can write it in your own words. Handing in a solution which coincides literally with another one will be considered cheating.

REFERENCES

1. E. Cerdá. Optimización dinámica, Prenctice Hall, 2001
2. S.E. Dreyfus, Dynamic programming and the calculus of variations, Academic Press, 1965.
3. Nelson Dunford and Jacob T. Schwartz, Linear operators, John Wiley & Sons, 1988
4. Gerald B. Folland, Real analysis: modern techniques and their applications, Jhon Wiley & Sons, [1999]
5. I.M. Gelfand, S.V. Fomin, Calculus of variations, Prentice-Hall, 1963
6. G. Hadley, M.C. Kemp, Variational methods in economic, North-Holland, 1973
7. Michael D. Intriligator, Mathematical optimization and economic theory, Prentice-Hall, cop, 1971
8. I. kamien and Nancy L. Schawartz, Dynamic optimization: the calculus of variations and optimal control in economics and management, North-Holland, cop.1981
9. A.N, Kolmogorov and S.V. Fomin, Elements of the theory of functions and functional analysis, vo, 1, Graylock Press, 1957-1961
10. A.N. Lolmogorov and S.V. Fomin, Introductory real analysis, New York: Dover, 1975
11. Daniel Leonard, ngo van Long, Optimal control theory and statc optimization in economics, Cambridge University Press, 1992
12. D. G. Luenberger, Optimization by Vector Space Methods, Wiley 1997
13. Magnus R. Hestenes, Calculus of variations and optimal control theory, John Wiley & Sons, cop.1966
 
 

Comentarios y sugerencias: Luis Miguel Sánchez Sánchez - luismi@pa.uc3m.es
Última actualización: 17 de enero de 2002