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Spring 2021 - Problem 5

uniform boundedness principle

Let $x \in \R^\N$ be such that the series

\sum_{i=1}^\infty x_iy_i

converges for all $y \in \R^\N$ such that $\lim_{n\to\infty} y_n = 0$ . Show that the series $\sum_{n=1}^\infty \abs{x_n}$ converges.

Solution.

Consider $\set{y \mapsto \sum_{i=1}^n x_iy_i}_n$ , which is a family of linear functionals on $c_0\p{\N}$ . Since each sum is finite, it's clear that each functional is bounded. Moreover, for every $y \in c_0\p{\N}$ , notice that $\set{\sgn\p{x_i} y_i}_i \in c_0\p{\N}$ also, and so

\sup_n \,\abs{\sum_{i=1}^n x_iy_i} \leq \sum_{i=1}^\infty \abs{x_iy_i} = \sum_{i=1}^\infty x_i \p{\sgn\p{x_i}y_i} < \infty,

by assumption. Thus, by the uniform boundedness principle, we see that

\sum_{i=1}^\infty \abs{x_i} = \sup_n \,\sum_{i=1}^n \abs{x_i} = \sup_n \,\norm{y \mapsto \sum_{i=1}^n x_iy_i}_{c_0\p{\N}\to\R} < \infty,

which was what we wanted to show.