Course Name Code Semester T+U Hours Credit ECTS
Topological Vector Spaces-II MAT 508 0 3 + 0 3 6
Precondition Courses Topological Vector Spaces-I
Recommended Optional Courses
Course Language Turkish
Course Level yuksek_lisans
Course Type Optional
Course Coordinator Doç.Dr. MAHPEYKER ÖZTÜRK
Course Lecturers
Course Assistants
Course Category Available Basic Education in the Field
Course Objective To apprehend the properties of linear transformations,to learn the concept of dual, general open mapping and closed graph theorems, tensor products and nuclear spaces,absolute summability, weak compactness,to understand the Eberlein and Krein theorems.
Course Content Linear transformations(continuous linear transformations and topological homomorphism,Banach Homomorphism Theorem,Linear transformation spaces,equcontinuity,the principle of uniform boundedness and Banach-Steinhaus theorem,Bilinear transformations,topological tensor products,nuclear transformations and spaces,examples,approximation,compact transformations).The concept of dual(dual systems and weak topologies, the fundamental properties of adjoint transformations,the compatible local convex topologies with a given dual,Mackey-Arens theorem,Projective duals and reduction topologies,The strong dual of local convex spaces,bidual,reflexive spaces,the dual characterization of completeness,metrizable spaces,the adjoint of the closed linear transformations,general open mapping and closed graph theorems,tensor products and nuclear spaces,absolute summability,weak compactness,Eberlein and Krein theorems
# Course Learning Outcomes Teaching Methods Assessment Methods
1 He/she recognizes the continuous linear transformations and topological homomorphism, Banach Homomorphism Theorem, Space of linear transformations and the concept of equcontinuity. Lecture, Question-Answer, Discussion, Self Study, Testing, Homework,
2 He/she expresses and proves the principle of uniform boundedness and Banach-Steinhaus theorem, Lecture, Question-Answer, Discussion, Self Study, Testing, Homework,
3 He/she explains Bilinear transformations,topological tensor products, nuclear transformations and spaces,examples,approximation problem and compact transformations. Lecture, Question-Answer, Discussion, Self Study, Testing, Homework,
4 He/she expresses the Dual systems and weak topologies, main properties of adjoint transformations Lecture, Question-Answer, Discussion, Self Study, Testing, Homework,
5 He/she expresses the General open mapping and closed graph theorems. Lecture, Question-Answer, Discussion, Self Study, Testing, Homework,
6 He/she interprets the Tensor products and nuclear spaces. Lecture, Question-Answer, Discussion, Self Study, Testing, Homework,
7 He/she interprets the absolute summability,weak compactness,Eberlein and Krein theorems. Lecture, Question-Answer, Discussion, Self Study, Testing, Homework,
Week Course Topics Preliminary Preparation
1 continuous linear transformations and topological homomorphism
2 Banach Homomorphism Theorem,Linear transformation spaces
3 equcontinuity,the principle of uniform boundedness and Banach-Steinhaus theorem,
4 Bilinear transformations
5 topological tensor products,nuclear transformations and spaces
6 approximation,compact transformations
7 dual systems and weak topologies
8 the fundamental properties of adjoint transformations,the compatible local convex topologies with a given dual
9 intermediate examination
10 Mackey-Arens theorem,Projective duals , reduction topologies
11 The strong dual of local convex spaces,bidual,reflexive spaces,the dual characterization of completeness,
12 general open mapping and closed graph theorems
13 tensor products and nuclear spaces,
14 weak compactness,Eberlein and Krein theorems
Resources
Course Notes [1] Musayev, Binali; Fonksiyonel Analiz, Balcı Yayınları, 2000, İstanbul
Course Resources [2] Maddox,I.J.; Elements of Functional Analysis, Cambridge Un.Press,1970,London.
[3] Şuhubi, Erdoğan; Fonksiyonel Analiz, İTÜ Vakfı, 2001, İstanbul
[4] Naylor, Arch; Linear Operator Theory in Engineering and Science, Springer-Verlag, 1982.
Order Program Outcomes Level of Contribution
1 2 3 4 5
2 Student follows the current journals in his/her field and puts forward problems. X
2 Student follows the current journals in his/her field and puts forward problems. X
3 Student understands the relations between the disciplines pertaining to the undergraduate programs of Mathematics X
3 Student understands the relations between the disciplines pertaining to the undergraduate programs of Mathematics X
4 Student gets new knowledge by relating the already acquired experience and knowledge with the subject-matters out of his/her field. X
4 Student gets new knowledge by relating the already acquired experience and knowledge with the subject-matters out of his/her field. X
5 Student uses different proof methods to come to a solution by analyzing the problems encountered. X
5 Student uses different proof methods to come to a solution by analyzing the problems encountered. X
6 Student determines the problems to be solved within his/her field and if necessary takes the lead. X
6 Student determines the problems to be solved within his/her field and if necessary takes the lead. X
7 Student conveys, in team work, his/her knowledge in the studies done in different disciplines by applying the dynamics pertaining to his/her own field. X
7 Student conveys, in team work, his/her knowledge in the studies done in different disciplines by applying the dynamics pertaining to his/her own field. X
8 Student critically evaluates the knowledge got at the bachelor´s degree level, makes up the missing knowledge and focuses on the current subject-matters X
8 Student critically evaluates the knowledge got at the bachelor´s degree level, makes up the missing knowledge and focuses on the current subject-matters X
9 Student knows a foreign language to communicate orally and in writing and uses the foreign language in a way that he/she can have a command of the Maths terminology and can do a source research. X
9 Student knows a foreign language to communicate orally and in writing and uses the foreign language in a way that he/she can have a command of the Maths terminology and can do a source research. X
10 Student improves himself/herself at a level of expertness in Mathematics or in the fields of application by improving the knowledge got at the bachelor´s degree level. X
10 Student improves himself/herself at a level of expertness in Mathematics or in the fields of application by improving the knowledge got at the bachelor´s degree level. X
11 Student considers the scientific and cultural ethical values in the phases of gathering and conveying data or writing articles. X
Evaluation System
Semester Studies Contribution Rate
1. Ara Sınav 80
1. Ödev 10
2. Ödev 10
Total 100
1. Yıl İçinin Başarıya 50
1. Final 50
Total 100
ECTS - Workload Activity Quantity Time (Hours) Total Workload (Hours)
Course Duration (Including the exam week: 16x Total course hours) 16 3 48
Hours for off-the-classroom study (Pre-study, practice) 16 3 48
Mid-terms 1 20 20
Assignment 2 10 20
Final examination 1 25 25
Total Workload 161
Total Workload / 25 (Hours) 6.44
dersAKTSKredisi 6