Fundamental Algebra and Analysis I

Prerequisites:

None

Course description:

This course is designed for first year students who intend to pursue studies in fundamental mathematics or applied mathematics major. It is also open to all those students who are interested in modern mathematics. The teaching objective is to enable students to master the fundamental knowledge and language of algebra and analysis, in order to prepare them for in-depth study of modern mathematics. Topics focus on fundamental mathematical structures—including sets, functions, algebraic structures, order structures, topological structures, etc., and together with differential and integral calculus. This course will employ a variety of teaching methods, including lectures, class discussions, problem-solving sessions, and independent reading.

Learning objectives:

By the end of this course, students will be able to:

Analyze fundamental mathematical structures—including sets, functions, algebraic systems (groups, rings, fields), and ordered sets—applying key theorems such as the Knaster-Tarski and Cantor-Bernstein theorems.

Work with real and extended real numbers, proving properties from axioms, evaluating limits of sequences and series, and applying convergence theorems (monotone, Bolzano-Weierstrass, dominated convergence).

Demonstrate proficiency in linear algebra by constructing and manipulating modules, submodules, and linear mappings, solving systems of equations, and utilizing matrix representations.

Apply differential calculus in normed spaces, including differentiation rules, Taylor expansions, and optimization, while understanding topological and metric properties.

Compute and manipulate integrals, using techniques such as integration by parts, change of variables, and parameter-dependent integrals, while applying convergence theorems.

Synthesize abstract concepts across algebra, analysis, and linear algebra to solve theoretical and applied mathematical problems.

Detailed topics covered:

1. Basic logic: mathematical statements, negation, conjunction and disjunction, conditional statement, biconditional statement, proof by contraposition, proof by contradiction.

2. Sets: roster notation, subset and power set, set-builder notation, set difference, quantifiers, sufficient and necessary condition, union, intersection

3. Correspondences: correspondences between two sets, inverse correspondence, image and preimage, composition, surjectivity, injectivity, mappings, bijection, direct product, restriction and extension, equivalence relation

4. Ordering: partially ordered sets, monotonic functions, supremum and infimum, intervals, order-completeness, mathematical induction, finiteness and countability, Zorn’s lemma.

5. Groups: composition laws, neutral element, invertible elements, substructure, homomorphisms, quotient structure, universal homomorphism, product, action of monoids on a set.

6. Rings and modules: unitary ring, homomorphism, modules over a unitary ring, submodules, module homomorphism, matrices, linear equations, quotient modules, quotient ring, free modules, rank, algebra, formal series.

7. Limits: filters, order limits, partially ordered semigroups, enhancement, extrema in a partially ordered set, absolute value, ordered rings, enhancement of a totally ordered ring.

8. Topological spaces: topology, convergence in a topological space, metric space, continuity, initial topology, uniform continuity, closed subsets, completeness, compactness, compact metric spaces, path connectedness.

9. Differential calculus: Landau symbol, differentiability, convexity, mean value property, multilinear mappings, bounded multilinear mappings, higher differential, Taylor formula, series in a Banach space, local inversion, uniform convergence, power series, directional differential

10. Integral calculus: differential 1-forms, primitive functions, Daniell integral, set semirings, sigma-additive functions, measurable space, integration on a measurable space, product measure.