Course Unit

Sets, operations with sets, subsets, power set, Cartesian product. Sets of numbers, integers, rational numbers, gentle introduction to real numbers.

A resume of the algebraic properties of rational numbers, ordering in rational numbers. Definition of real numbers through decimal alignment. Absolute value. Intervals. A non-rigorous definition of the separation property or real numbers.

Introduction to functions on arbitrary sets. Image and pre-image. Surjective and injective functions. Composition of functions. The identical function on a set. Invertible functions and their inverse.

Power laws, exponentials, and logarithms in the set of real numbers. 

Cardinality of infinite sets. Countable sets. Cardinality of real numbers.

More on functions of real numbers. Domain of a function. Examples. Operations on the graph of a functions through translations, dilations, and absolute values. Examples of elementary functions. Bounded functions, monotone functions, even and odd functions, periodic functions. Trigonometric functions and their inverse.

Upper bounds and lower bounds. Bounded and unbounded subsets of the real line. Maxima and minima. Supremum and infimum. Examples.

Complex numbers. Cartesian form, real and imaginary part, conjugacy, modulus, operations with complex numbers. Representation on the complex plane. Trigonometric form. De Moivres' formulas. Roots of a complex number. Algebraic equations on complex numbers. Fundamental theorem of algebra.

Polynomials on real numbers. Algebraic and transcendental equations. Trigonometric equations, exponential equations, logarithmic equations. Exercises.

The Cartesian plane. Geometric loci on the plane: straight lines, parabolas, circles, ellipses, hyperboles. Exercises. Solution of nonlinear systems and intersection of geometric loci in the plane.

Solution of algebraic and transcendental inequalities on the real line. Exercises.

Introduction to the study of the graph of a functions of real variables. Domain, zeroes, sign. Exercises

Propositional logic. Propositional calculus.

Sets, set operations, relations, functions. Cardinality of sets, countable sets, uncountable sets. Elementary number sets. Integers and rationals. Induction principle.

More on functions: injective and surjective functions, invertible functions, image and pre-image.

The set of real numbers. Separation axiom, Dedekind cuts. Infimum and supremum. Archimedean property.  Complex numbers: cartesian and trigonetric form, basic properties, powers, complex roots, fundamental theorem of algebra.

Sequences of real numbers: monotone sequences, convergence of a sequence, subsequences, limsup and liminf of a sequence, Bolzano-Weierstrass theorem.

Introduction to functions of real numbers. Elementary functions: exponential and logarithmic function, trigonometric functions, irrational functions. Monotone functions.

The topology of real numbers: intervals, half lines, open sets, closed sets. The topology of the Euclidean space Rn: balls, open and closed sets. Compact sets in the Euclidean space.

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
• Introduction to the theory of unbounded operators. Linear differential operators. Applications.
• Introduction to infinite-dimensional differential calculus and variational methods.

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on abstract measure and integration. Measurable spaces and measurable functions. Borel and Lebesgue measures. Integrals on measure spaces. Limit exchange convergence theorems. Lp spaces. Product measures. Signed measures and Radon-Nicodym Theorem. Riesz representation theorem for measures.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.