Subject: Functional Analysis

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - Identification and realization of complete metric spaces. Identification the completion of a incomplete metric or normed spaces
2 - Identification Hilbert spaces. Geometric properties, Total sets
3 - Linear operators: Identification and construction. Dual spaces: Identification and construction

# Syllabus:

1 - Metric spaces: Recall
1.1 - Complete metric spaces
1.2 - Complete metric spaces and completion
1.3 - Continuous maps between metric spaces
2 - Normed spaces and Banach spaces
2.1 - Definition and examples of normed and Banach spaces
2.2 - Schauder basis for separable normed spaces
2.3 - Finite dimensional normed spaces
3 - Hilbert spaces
3.1 - Inner product, space with inner product and Hilbert spaces. Examples
3.2 - Orthogonality
3.3 - Classic (in)equalities, Cauchy-Schwarz, parallelogram and polarization. Isomorphic Hilbert spaces
4 - Linear operators
4.1 - Linear operators between normed spaces. Examples
4.2 - Inverse operator
4.3 - Bounded and continuous linear operator
4.4 - Norm of a bounded linear operator
4.5 - Restriction and extension of a bounded linear operator
5 - Linear functionals
5.1 - Bounded linear functional. Linear functionals in finite dimensional spaces
5.2 - The space of linear operators as normed space
5.3 - Dual space of a normed space

# Literature/Sources:

E. Kreyszig , 1978 , Introductory Functional Analysis With Applications , John Wiley & Sons
M. Reed B. Simon , 1975 , Methods of Modern Mathematical Physics , Elsevier
Y. M. Berezansky, Z. Sheftel, G. F. Us , 1996 , Functional Analysis , Birkhäuser
B.P. Rynne, M.A. Youngson , 2008 , Linear Functional Analysis , Springer London
H. W. Alt , 2016 , Linear Functional Analysis: An Application-Oriented Introduction , Springer

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
The character of this unit is namely theoretical. We follow the syllabus using examples and exercises with a proper level of difficulty. Interesting problems are proposed during the semester as a complement activity. Evaluation model A, each of the two evaluation weights 50% of the final mark. In the appeal exam it is possible to recover 100%.