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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 4058 TOPOLOGICAL VECTOR SPACES ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSISTANT PROFESSOR CELAL CEM SARIOĞLU

Offered to

Mathematics (Evening) Mathematics

Course Objective

This course introduces students to the idea of thinking the two concepts topology and vector spaces together.

Learning Outcomes of the Course Unit

1   Will be able to express vector spaces as topological spaces 2   Will be able to know properties of convex sets 3   Will be able to separate convex sets 4   Will be able to know Hahn-Banach theorem and its consequences 5   Will be able to know the L_p-spaces and their properties

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Normed vector spaces, Norm isomorphism, Banach spaces 2 Quotient spaces and topological product, The dual space, Continuous linear mappings 3 The spaces c_0, c and l_1, The spaces l_p 4 The (B)-spaces of holomorphic and continuous functions, The spaces L_p (p>1) . 5 Definitions of topological vector spaces, Completion 6 Quotient spaces and topological products, Finite dimensional topological vector spaces 7 Bounded and compact subsets, locally compact topological spaces, Topologically complementary spaces 8 Midterm 9 The dual space, hyperplanes, the spaces L_p with 0 10 Metrizable spaces, The Banach-Schauder theorem and the closed graph theorem, Equicontinuous mappings, The theorems of Banach and Banach-Steinhaus, Bilinear mappings 11 The convex and absolutely convex cover of a set, The algebraic boundary of a convex sets, Half spaces 12 Convex bodies and Minkowski functional, Covexcones, Hypercones 13 Seperation theorem, The Hahn-Banach theorem, The analytic proof of Hahn-Banach theorem and its two consequences 14 Supporting hyperplanes, The Hahn-Banch theorem for Normed vector spaces, Adjoint mappings, The dual space of C(I)

Recomended or Required Reading

Textbooks: 1. Köthe, G., Topological Vector Spaces I, Springer, 1969, ISBN 978-0387045092 2. Lawrence, N., Topological Vector Spaces, Springer, 1985, ISBN 0824773152 Materials: Özçelik, Ahmet Z., Topoloji Ders Notları

Planned Learning Activities and Teaching Methods

Lectures, Lecture notes

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 ASG ASSIGNMENT 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE MTE * 0.40 + ASG * 0.10 + FIN * 0.50 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.40 + ASG * 0.10 + RST * 0.50 *** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

E-mail: ahmet.ozcelik@deu.edu.tr Office : 0 232 3018585

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 4 52 Preparations before/after weekly lectures 12 3 36 Preparation for midterm exam 1 25 25 Preparation for final exam 1 35 35 Preparing assignments 4 4 16 Final 1 2,5 3 Midterm 1 2,5 3 TOTAL WORKLOAD (hours) 170

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 PO.12 PO.13 LO.1 4 4 3 LO.2 4 4 3 LO.3 4 4 4 3 3 LO.4 4 3 3 LO.5 4 4 3