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

## 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}_{→∞}(2

*x*

_{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.

## 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

## Additional Details

Monday, March 18, 2019 at 3:00 p.m. in Math 103

Refreshments at 4:00 p.m. in Math Lounge 109