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Functional Analysis Lecture Notes - Michigan State University

Functional Analysis (Math 920) Lecture Notes for Spring 08 Jeff SchenkerMichigan State UniversityE-mail and course informationPart 1. Hahn-Banach Theorem and ApplicationsLecture 1. Linear spaces and the Hahn Banach TheoremLecture 2. Geometric Hahn-Banach TheoremsLecture 3. Applications of Hahn-BanachPart 2. Banach SpacesLecture 4. Normed and Banach SpacesLecture 5. Noncompactness of the Ball and Uniform ConvexityLecture 6. Linear Functionals on a Banach SpaceLecture 7. Isometries of a Banach SpaceHomework IPart 3. Hilbert Spaces and ApplicationsLecture 8. Scalar Products and Hilbert SpacesLecture 9. Riesz-Frechet and Lax-Milgram TheoremsLecture 10. Geometry of a Hilbert space and Gram-Schmidt processPart 4. Locally Convex SpacesLecture 11. Locally Convex Spaces and Spaces of Test FunctionsLecture 12. Generation of a LCS by seminorms and Fr echet SpacesLecture 13. The dual of an LCSL ecture 14. Spaces of distributionsLecture 15.

Reading: x3.2, x3.3 of Lax. We may use Hahn-Banach to understand something of the geometry of linear spaces. We want to understand if the following picture holds in in nite dimension: Figure 2.1. Separating a point from a convex set by a line hyperplane Definition 2.1. A set S ˆX is convex if for all x;y 2S and t 2[0;1] we have tx+ (1 t)y2S.

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