# Analysis Seminar

#### Analysis of Concentration in Banach Spaces

**Speaker:**
Cyril Tintarev, Uppsala University and University of California, Irvine (emeritus)

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, April 23, 2015, 2 p.m.

**Synopsis:**

Concentration compactness principle says that when compactness of an imbedding between two normed vector spaces is lost due to invariance of the norms with respect to action of a non-compact group, it can be partially restored by accounting to "blowups" by the group action. The formal expression of it is the profile decomposition: every bounded sequence has a subsequence that converges (in a stronger than weak topology) up to a sum of the blowup terms. We give a general profile decomposition result for uniformly convex Banach spaces.

Surprisingly, the underlying convergence is not the weak convergence, but Delta-convergence of T.-C.Lim, known to experts in the fixed point theory, and related to the notion of asymptotic centers. In Hilbert spaces Delta convergence and weak convergence coincide. We discuss properties of Delta-convergence (including Delta-compactness) and its relation to weak convergence. We also give an analog of Brezis-Lieb Lemma where the condition of a.e. convergence is replaced by Delta- and weak convergence to the same limit. This is a joint work with Sergio Solimini.