Functional Analysis Seminar 2014

functional analysis seminar - spring 2014

subject

The functional analysis seminar in Spring 2014 will be on geometry of Banach spaces.

Banach spaces can be seen as a generalization of Hilbert spaces. In Hilbert space theory, orthonormal bases and orthogonal projections play an important role. Are there similar structures in Banach spaces? Banach space theory has considered such questions already since its initiation by Banach and his co-workers. It turns out that the answers to many questions depend on properties of the norm of the Banach space under consideration, or, stated differently, on the shape of its unit ball. Decades of research have created a rich branch of functional analysis: geometry of Banach spaces.

We will first consider bases in Banach spaces, which are defined to be sequences such that every element can be written uniquely as an `infinite linear combination' of the basis elements. There are issues concerning convergence of such series. We will also study coordinate maps and coordinate projections related to the bases. There is a surprisingly detailed theory and there are amazing examples.

As a second topic we will consider properties of the unit ball. In particular we will study strict convexity properties and several types of smoothness. It turns out that such concepts are dual: certain convexity properties of the unit ball are equivalent to certain smoothness properties of the unit ball of the dual space. As an application we may look into a characterization of the isometries on the sequence spaces l_p with p between 1 and infinity and not equal to 2, or a characterization of ranges of contractive projections in terms of the duality map in reflexive strictly convex Banach spaces with strictly convex dual spaces.

book

We will follow Chapters 4 and 5 of the book:

Robert E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics Vol. 183, Springer Verlag, 1998.

This book is available from Springer for just over 70 euro or from most major bookshops. You might try to buy a considerably cheaper international edition at AbeBooks or similar websites.

We may study some additional handouts as well, e.g.,

Bruce Calvert, Convergence sets in reflexive Banach spaces, Proc.Amer.Math.Soc.47 (1975), 423--428.

prerequisites

Participation in the seminar requires an introductory course on functional analysis as a prerequisite, including most of the following topics: duals of Banach spaces, Hahn-Banach theorem, uniform boundedness principle, weak and weak* topologies, equivalence of norms, isometries, Hilbert spaces, orthonormal bases, orthogonal projections, Cauchy-Schwarz inequality, the spaces of p-summable sequences, Holder's inequality, the space of sequences converging to zero. The mastermath course Functional Analysis is not a prerequisite.

time

We will meet on Fridays from 14.00-17.00 in the Snellius building, Leiden University. This is the (still tentative) program:

February 14.

February 21.

February 28.

March 7.

March 21.

March 28.

April 4.

April 11.

April 25.

May 2.

May 9.

May 16.

May 23.

participation

By active participation it is possible to obtain 6 EC for this seminar. Active participation includes attending (almost) all sessions and giving a one afternoon lecture on an assigned part of the book. Please inquire at your own university if it is allowed to include this seminar in your master program. Some more formal details can be found in the study guide.

If you have further questions or if you like to participate, please contact Onno van Gaans.