# Dr. Sven-Ake Wegner, Universität Wuppertal: "The heart of the Banach spaces"

Feb 02, 2017 from 02:30 PM to 03:30 PM

LMS 4 - Raum 424 - Kleiner Hörsaal

Abstract:

Let $X$ be a Banach space and let $Y\subseteq X$ be a linear subspace. If $Y$ is closed in $X$ then $X/Y$ is a Banach space in the quotient norm. If $Y$ is not closed then this wrong---even if $Y$ is a Banach space in a norm stronger than those induced by $X$. Examples are the inclusions $\ell^{1}\subseteq c_0$ or $C^1[0,1]\subseteq C[0,1]$.

The unpleasant fact, that there is no reasonable Banach space $X/Y$ in the setting above, motivated Waelbroeck in the 1960s to consider formal quotients of Banach spaces. Amazingly, in 1982, the same year in which he published his paper on the category of quotient Banach spaces, Be{\u\i}linson, Bernstein and Deligne published in a geometric context a very abstract and by now very famous theory about hearts of t-structures on triangulated categories. It turnes out that in their terminology, and for the special case of Banach spaces, the heart is precisely the category of formal quotients considered by Waelbroeck.

In the talk we sketch the definition of the heart and discuss possible generalizations beyond the case of Banach spaces.