Code MATH 402
Term 201901
Title Hilbert Space Techniques
Faculty Faculty of Engineering and Natural Sciences
Subject Mathematics(MATH)
SU Credit 3
ECTS Credit 6.00 / 6.00 ECTS (for students admitted in the 2013-14 Academic Year or following years)
Instructor(s) Albert Erkip albert@sabanciuniv.edu,
Detailed Syllabus
Language of Instruction English
Prerequisites
(only for SU students)
--
Mode of Delivery Formal lecture
Content

Inner product, Hilbert space, examples, orthogonal expansions. Classical Fourier series; The Fejer kernel, Fejer's theorem, Parseval's formula, Weierstrass approximation theorem. Dual space, the Riesz-Frechet theorem. Linear operators, multiplication operators and infinite operator matrices, compact Hermitian and Hibert-Schmidt operators and the spectral theorem. Applications.

Objective

To introduce inner product spaces and its examples. Present orthogonal

expansions, Classical Fourier series, the Fejer kernel, Fejer's theorem,

Parseval's formula and Weierstrass approximation theorem. Provide a basic

understanding of dual space, the Riesz-Frechet theorem. Define linear operators,

multiplication operators and infinite operator matrices, compact Hermitian

and Hilbert-Schimidt operators. Prove the spectral theorem and articulate its

applications in the context of functional analysis.

Learning Outcome

Upon completing this course students should be able to:
1. Describe the structure and properties of inner-product and normed space
2. Identify the concept and properties of a complete orthonormal sequence and compute the related Fourier Series expansions.
3. Apply and use the Riesz Representation Theorem
4. Describe continuity and relate it with bounded linear operators.
5. Apply the Closed Graph Theorem and its consequences.
6. Describe what is meant by the terms self-adjoint operator, normal operator, unitary operator and projection operator
7. Conceptualize and find the resolvent operator and find the spectrum of basic self-adjoint operators.
8. Apply and use Spectral Theorem and basic properties of eigenvalues and eigenvectors of bounded self-adjoint operators within the theory or Hilbert Spaces.

Programme Outcomes

 Common Outcomes For All Programs 1 Understand the world, their country, their society, as well as themselves and have awareness of ethical problems, social rights, values and responsibility to the self and to others. 1 2 Understand different disciplines from natural and social sciences to mathematics and art, and develop interdisciplinary approaches in thinking and practice. 5 3 Think critically, follow innovations and developments in science and technology, demonstrate personal and organizational entrepreneurship and engage in life-long learning in various subjects. 4 4 Communicate effectively in Turkish and English by oral, written, graphical and technological means. 1 5 Take individual and team responsibility, function effectively and respectively as an individual and a member or a leader of a team; and have the skills to work effectively in multi-disciplinary teams. 1 Common Outcomes ForFaculty of Eng. & Natural Sci. 1 Possess sufficient knowledge of mathematics, science and program-specific engineering topics; use theoretical and applied knowledge of these areas in complex engineering problems. 5 2 Identify, define, formulate and solve complex engineering problems; choose and apply suitable analysis and modeling methods for this purpose. 2 3 Develop, choose and use modern techniques and tools that are needed for analysis and solution of complex problems faced in engineering applications; possess knowledge of standards used in engineering applications; use information technologies effectively. 1 4 Ability to design a complex system, process, instrument or a product under realistic constraints and conditions, with the goal of fulfilling specified needs; apply modern design techniques for this purpose. 1 5 Design and conduct experiments, collect data, analyze and interpret the results to investigate complex engineering problems or program-specific research areas. 1 6 Knowledge of business practices such as project management, risk management and change management; awareness on innovation; knowledge of sustainable development. 1 7 Knowledge of impact of engineering solutions in a global, economic, environmental, health and societal context; knowledge of contemporary issues; awareness on legal outcomes of engineering solutions; understanding of professional and ethical responsibility. 1 Electronics Engineering Program Outcomes Area Electives 1 Use mathematics (including derivative and integral calculations, probability and statistics), basic sciences, computer and programming, and electronics engineering knowledge to design and analyze complex electronic circuits, instruments, software and electronics systems with hardware/software. 1 2 Analyze and design communication networks and systems, signal processing algorithms or software using advanced knowledge on differential equations, linear algebra, complex variables and discrete mathematics. 1 Materials Science and Nano Engineering Program Outcomes Area Electives 1 Applying fundamental and advanced knowledge of natural sciences as well as engineering principles to develop and design new materials and establish the relation between internal structure and physical properties using experimental, computational and theoretical tools. 1 2 Merging the existing knowledge on physical properties, design limits and fabrication methods in materials selection for a particular application or to resolve material performance related problems. 1 3 Predicting and understanding the behavior of a material under use in a specific environment knowing the internal structure or vice versa. 1 Mechatronics Engineering Program Outcomes Area Electives 1 Familiarity with concepts in statistics and optimization, knowledge in basic differential and integral calculus, linear algebra, differential equations, complex variables, multi-variable calculus, as well as physics and computer science, and ability to use this knowledge in modeling, design and analysis of complex dynamical systems containing hardware and software components. 1 2 Ability to work in design, implementation and integration of engineering applications, such as electronic, mechanical, electromechanical, control and computer systems that contain software and hardware components, including sensors, actuators and controllers. 1 Industrial Engineering Program Outcomes Area Electives 1 Formulate and analyze problems in complex manufacturing and service systems by comprehending and applying the basic tools of industrial engineering such as modeling and optimization, stochastics, statistics. 1 2 Design and develop appropriate analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy. 1 3 Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 1
 Assessment Methods and Criteria Percentage (%) Final 30 Midterm 30 Homework 40
 Recommended or Required Reading Textbook Introduction to Hilbert Space, Nicolas Young. (you will find it in our Information Center / reserve)