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# Functional Analysis I: Course Outline (MAT 411)

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.)

## Course Objective

At the end of this course the students will be able to understand a strong foundation in functional analysis, focusing on spaces (Metric Spaces, Normed Spaces, Inner Product Spaces) Operators, Fundamental Theorems and Applications. To strengthen students understanding of this theory through applications of functional analysis. To develop students skills and confidence in mathematical analysis and proof techniques. To build an understanding of mathematical analysis through the use of mathematical proof.

## Course Contents

• Metric spaces, dense subspaces, seperable spaces, Isometry, Definition and examples of normed spaces, Banach spaces, Characterization of Banach spaces,
• Bounded linear operators, Functionals and their examples, Various characterizations of bounded (continuous) linear operators, Space of all bounded linear operators,
• Open mapping and closed graph theorems, Dual (conjugate) spaces, Reflexive spaces, Hahn-Banach theorem (without proof), Some important consequences of the Hahn-Banach theorem.,
• Inner product spaces and their examples, Cauchy-Schwarz inequality, Hilbert spaces, Orthogonal complements, Projection theorem, Riesz representation theorem.