# Functional analysis;: An introduction (Pure and applied mathematics, v. 15)

Three or more years of high school mathematics or equivalent recommended. Review of polynomials. Graphing functions and relations: graphing rational functions, effects of linear changes of coordinates. Circular functions and right triangle trigonometry. Reinforcement of function concept: exponential, logarithmic, and trigonometric functions. Conic sections. Polar coordinates.

Differential calculus of functions of one variable, with applications. Functions, graphs, continuity, limits, derivatives, tangent lines, optimization problems. Integral calculus of functions of one variable, with applications. Antiderivatives, definite integrals, the Fundamental Theorem of Calculus, methods of integration, areas and volumes, separable differential equations. Introduction to functions of more than one variable. Vector geometry, partial derivatives, velocity and acceleration vectors, optimization problems.

No credit given if taken after or concurrent with 20C. Discrete and continuous random variables: mean, variance; binomial, Poisson distributions, normal, uniform, exponential distributions, central limit theorem. Sample statistics, confidence intervals, hypothesis testing, regression.

Introduction to software for probabilistic and statistical analysis. Emphasis on connections between probability and statistics, numerical results of real data, and techniques of data analysis. Basic discrete mathematical structure: sets, relations, functions, sequences, equivalence relations, partial orders, and number systems.

## Introduction to Functional Analysis

Methods of reasoning and proofs: propositional logic, predicate logic, induction, recursion, and pigeonhole principle. Infinite sets and diagonalization. Basic counting techniques; permutation and combinations. Applications will be given to digital logic design, elementary number theory, design of programs, and proofs of program correctness. Equivalent to CSE Matrix algebra, Gaussian elimination, determinants. Linear and affine subspaces, bases of Euclidean spaces. Eigenvalues and eigenvectors, quadratic forms, orthogonal matrices, diagonalization of symmetric matrices.

Computing symbolic and graphical solutions using Matlab. Students who have not completed listed prerequisites may enroll with consent of instructor.

### Notations and Assumptions

Foundations of differential and integral calculus of one variable. Functions, graphs, continuity, limits, derivative, tangent line. Applications with algebraic, exponential, logarithmic, and trigonometric functions. Introduction to the integral. Integral calculus of one variable and its applications, with exponential, logarithmic, hyperbolic, and trigonometric functions. Methods of integration. Infinite series. Polar coordinates in the plane and complex exponentials. Vector geometry, vector functions and their derivatives.

Partial differentiation. Maxima and minima.

Double integration. Ordinary differential equations: exact, separable, and linear; constant coefficients, undetermined coefficients, variations of parameters.

## Monographs and Surveys in Pure and Applied Mathematics

Series solutions. Laplace transforms. Techniques for engineering sciences. Vector fields, gradient fields, divergence, curl. Taylor series in several variables. Conservative fields. Topics include derivative in several variables, Jacobian matrices, extrema and constrained extrema, integration in several variables.

The Freshman Seminar Program is designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a small seminar setting. Freshman Seminars are offered in all campus departments and undergraduate colleges, and topics vary from quarter to quarter. Enrollment is limited to fifteen to twenty students, with preference given to entering freshman. Prerequisites: none. Cross-listed with EDS Students will develop skills in analytical thinking as they solve and present solutions to challenging mathematical problems in preparation for the William Lowell Putnam Mathematics Competition, a national undergraduate mathematics examination held each year.

Students must sit for at least one half of the Putnam exam given the first Saturday in December to receive a passing grade. May be taken for credit up to four times. Independent study or research under direction of a member of the faculty. First course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include groups, subgroups and factor groups, homomorphisms, rings, fields. Second course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include rings especially polynomial rings and ideals, unique factorization, fields; linear algebra from perspective of linear transformations on vector spaces, including inner product spaces, determinants, diagonalization.

ignamant.cl/wp-includes/55/3474-localizar-moviles-gsm.php Third course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include linear transformations, including Jordan canonical form and rational canonical form; Galois theory, including the insolvability of the quintic. Second course in linear algebra from a computational yet geometric point of view. Moore-Penrose generalized inverse and least square problems.

Difference Between Pure & Applied Mathematics - English

Vector and matrix norms. Characteristic and singular values. Canonical forms.