Functional Analysis (MAE719): Διαφορά μεταξύ των αναθεωρήσεων

Τελευταία αναθεώρηση της 18:45, 17 Αυγούστου 2024

Ελληνική Έκδοση
Undergraduate Courses Outlines
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Department of Mathematics
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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE719
Semester	7
Course Title	Functional Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes ( in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The goal of this course is: To familiarize the student with the notions, the basic theorems and the techniques concerning normed vector spaces, Banach spaces, Hilbert spaces, bounded linear operators between them and the dual spaces. After completing this course the student will be able to recognize if a given normed linear space is a Banach space, to compute the norm of a bounded linear operator, to use the basic theorems of Functional analysis (Hahn-Banach theorem and its consequences, Open mapping theorem, Closed graph theorem, Banach-Steinhaus theorem, Uniform Boundedness Principle).
General Competences	Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making.

Syllabus

Linear spaces and algebraic bases (Hamel bases). Linear operators. Normed linear spaces. Banach spaces and classical examples. Finite dimensional spaces. Bounded linear operators, bounded linear functionals and computation of their norm. Dual space. Conjugate operators. Hahn Banach theorem and its consequences. The second dual space. Reflexive spaces. Baire's category theorem and some of its consequences in Functional Analysis (Open Mapping Theorem, Closed graph Theorem, Uniform Boundedness Principle, Banach-Steinhaus Theorem). Elements from Hilbert spaces.

Teaching and Learning Methods - Evaluation

Delivery

Teaching on the blackboard from the teacher

Use of Information and Communications Technology

-

Teaching Methods

Activity	Semester Workload
Lectures	39
Individual study	78
Solving exercises-homework	33
Course total	150

Student Performance Evaluation

Exams in the end of the semester (mandatory), intermediate exams (optional), assignments of exercises during the semester (optional).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

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-[[Undergraduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
+* [[Συναρτησιακή Ανάλυση (ΜΑΕ719)|Ελληνική Έκδοση]]
+{{Course-UnderGraduate-Top-EN}}
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 === General ===
@@ Γραμμή 18: / Γραμμή 20: @@
 ! Course Code
 |
-MAE711
+MAE719
 |-
 ! Semester
-|
+| 7
 |-
 ! Course Title
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-Functional Analysis I
+Functional Analysis
 |-
 ! Independent Teaching Activities
@@ Γραμμή 48: / Γραμμή 49: @@
 |-
 ! Course Website (URL)
-| -
+| See [https://ecourse.uoi.gr/ eCourse], the Learning Management System maintained by the University of Ioannina.
 |}
 === Learning Outcomes ===
 {| class="wikitable"
@@ Γραμμή 55: / Γραμμή 57: @@
 ! Learning outcomes
 |
-The goal of this course is: To familiarize the student with the notions the basic theorems and the techniques concerning Banach spaces, bounded linear operators between them, dual spaces and especially Hilbert spaces.
+The goal of this course is: To familiarize the student with the notions, the basic theorems and the techniques concerning normed vector spaces, Banach spaces, Hilbert spaces, bounded linear operators between them and the dual spaces. After completing this course the student will be able to recognize if a given normed linear space is a Banach space, to compute the norm of a bounded linear operator, to use the basic theorems of Functional analysis (Hahn-Banach theorem and its consequences, Open mapping theorem, Closed graph theorem, Banach-Steinhaus theorem, Uniform Boundedness Principle).
-After completing this course the student will be able to recognize if a given normed linear space is a Banach space, to compute the norm of a bounded linear operator, will be able to use the basic theorems of Functional analysis (Hahn-Banach  theorem and its consequences, Open mapping theorem, Uniform Boundedness Principle), and will get the basic theorems and techniques concerning Hilbert spaces (e.g. existence of orthonormal bases, Gram-Schmidt orthogonalization procedure, isometry of every Hilbert space with its dual).
 |-
 ! General Competences
 |
-This course aims to provide the student with the theoretical background and the fluency of using the basic theorems and techniques of Functional Analysis. Promotes the analytical and synthetic thinking that the student will be able to apply the knowledge acquired in a broader scope including the whole range of mathematical analysis.
+* Αnalyse and combine data and information using various technologies.
+* Working independently and in groups.
+* Free, creative, analytic, and conclusive thinking.
+* Decision making.
 |}
 === Syllabus ===
-Linear spaces and algebraic bases (Hamel bases), linear operators. Normed spaces, Banach spaces, classical examples. Bounded linear operators, dual spaces, conjugate operators. Hahn Banach theorem and its consequences. Reflexive spaces. Inner product spaces, Hilbert spaces, orthonormal systems, every Hilbert space is isometric with its dual. Baire's category theorem and some of its consequences in Functional Analysis (Open Mapping Theorem, Closed graph Theorem, Uniform Boundedness Principle, Banach Steinhauss Theorem).
+Linear spaces and algebraic bases (Hamel bases). Linear operators. Normed linear spaces. Banach spaces and classical examples. Finite dimensional spaces. Bounded linear operators, bounded linear functionals and computation of their norm. Dual space. Conjugate operators. Hahn Banach theorem and its consequences. The second dual space. Reflexive spaces. Baire's category theorem and some of its consequences in Functional Analysis (Open Mapping Theorem, Closed graph Theorem, Uniform Boundedness Principle, Banach-Steinhaus Theorem). Elements from Hilbert spaces.
 === Teaching and Learning Methods - Evaluation ===
 {| class="wikitable"
@@ Γραμμή 98: / Γραμμή 105: @@
 |}
 === Attached Bibliography ===
-* Γενική Τοπολογία και Συναρτησιακή Ανάλυση,  Σ.  Νεγρεπόντης, Θ. Ζαχαριάδης, Ν. Καλαμίδας, Β. Φαρμάκη, Εκδόσεις Συμμετρία,  (κωδικός στο σύστημα Εύδοξος:  45321).
-* Στοιχεία Συναρτησιακής Ανάλυσης, Χ. Καρυοφύλλης, Εκδόσεις Ζήτη (κωδικός στο σύστημα Εύδοξος:  11278).
+<!-- In order to edit the bibliography, visit the webpage -->
-* Συναρτησιακή Ανάλυση, Haim Brezis, Εκδόσεις Ε.Μ.Π. (κωδικός στο σύστημα Εύδοξος:  20956).
+<!-- https://wiki.math.uoi.gr/index.php/%CE%A0%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF:MAE851-Biblio -->
+See the official [https://service.eudoxus.gr/public/departments#20 Eudoxus site] or the [https://cloud.math.uoi.gr/index.php/s/62t8WPCwEXJK7oL local repository] of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:
+{{MAE851-Biblio}}