2023
Topics of Functional Analysis
Name: Topics of Functional Analysis
Code: MAT14364M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese, English
Regime de Frequência: Presencial
Presentation
Functional Analysis is a branch of Mathematics that originated from classical Analysis and deals with the study of infinite dimensional function spaces. The UC intends to deepen and consolidate the knowledge of Functional Analysis and to show its applications to various areas of Analysis.
Sustainable Development Goals
Learning Goals
Objectives:
- To extend the knowledge of some subjects of Functional Analysis and analitic methods.
- To acquire skills to apply these methods in various fields of Analysis.
Competencies:
- Develop abstract thought as a means of solving, with both greater generality and simplicity, specific problems of Analysis.
- Abstraction skills, creative intuition, model construction and spirit of criticism.
- Skills for explaining the obtained results, both orally and written.
- To extend the knowledge of some subjects of Functional Analysis and analitic methods.
- To acquire skills to apply these methods in various fields of Analysis.
Competencies:
- Develop abstract thought as a means of solving, with both greater generality and simplicity, specific problems of Analysis.
- Abstraction skills, creative intuition, model construction and spirit of criticism.
- Skills for explaining the obtained results, both orally and written.
Contents
- Metric spaces. Convergence, Cauchy Sequences, Completeness. Examples and demonstrations of completeness. Examples of incomplete metric spaces. Completeness of spaces.
- Normed spaces. Banach spaces. Vectorial space. Normed space. Banach space. Finite dimensional spaces and subspaces. Compactness and finite dimension. Linear operators. Linear functionals. Operator normed spaces. Dual space.
- Hilbert spaces. Space with inner product. Hilbert space. Orthogonal complements and direct sums. Orthonormal sets and sequences. Orthonormal series and sets. Functionals in Hilbert spaces.
- Theorems in Banach spaces. Zorn's lemma. Hann-Banachs Theorem. Uniform Limitation Theorem. Strong and weak convergences
- Banach Fixed Point Theory. Banach Fixed Point Theorem. Banach's Theorem and Differential Equations. Banach's Theorem and Integral Equations.
- Normed spaces. Banach spaces. Vectorial space. Normed space. Banach space. Finite dimensional spaces and subspaces. Compactness and finite dimension. Linear operators. Linear functionals. Operator normed spaces. Dual space.
- Hilbert spaces. Space with inner product. Hilbert space. Orthogonal complements and direct sums. Orthonormal sets and sequences. Orthonormal series and sets. Functionals in Hilbert spaces.
- Theorems in Banach spaces. Zorn's lemma. Hann-Banachs Theorem. Uniform Limitation Theorem. Strong and weak convergences
- Banach Fixed Point Theory. Banach Fixed Point Theorem. Banach's Theorem and Differential Equations. Banach's Theorem and Integral Equations.
Teaching Methods
Classes are theoretical and practical, using a structured methodology for the presentation of the syllabus, supported by materials made available to students, and in examples of applications of the main results.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of the presentation of four works in which themes or problems related to the syllabus are addressed. Each work has a weight of 25% for the final classification;
2) Evaluation by exam, with the possibility of consulting material produced by the student.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of the presentation of four works in which themes or problems related to the syllabus are addressed. Each work has a weight of 25% for the final classification;
2) Evaluation by exam, with the possibility of consulting material produced by the student.
Teaching Staff
- Luís Manuel Balsa Bicho [responsible]
Certificações
