Generalizing a Real-Analysis Exam Problem: A Potpourri of Functional Analysis, Probability, and Topology

Authors

Leonard Huang, University of Nevada, Reno

Document Type

Presentation Abstract

Presentation Date

3-18-2019

Abstract

This talk is inspired by the following problem, which has tormented many a graduate student in real-analysis qualifying exams around the world:

Let (x_n)_n∈N be a sequence in R. If lim_n→∞(2x_n+1−x_n)=x for some x∈R, then prove that lim_n→∞ x_n =x also.

In the spirit of mathematical research, one may now ask: Is this result still true if we replace R by some other topological vector space? In this talk, we will show that the result is true for a wide class of topological vector spaces that includes all locally-convex ones, as well as some that are not locally convex, such as the L^p-spaces for p∈(0,1). We will then construct, using basic probability theory, an example of a badly-behaved topological vector space for which the result is false.

Additional Details

Monday, March 18, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Recommended Citation

Huang, Leonard, "Generalizing a Real-Analysis Exam Problem: A Potpourri of Functional Analysis, Probability, and Topology" (2019). Colloquia of the Department of Mathematical Sciences. 565.
https://scholarworks.umt.edu/mathcolloquia/565