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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2022 • Question 5

Question

Math 6210 • Real Analysis Spring 2022 • Question 5

Let \(V\), \(W\) be Banach spaces, and let \(T_{i} : V \to W\) be a sequence of bounded linear operators such that \(\lim _{i \to \infty } T_{i}v\) exists for every \(v \in V\). Define the linear map \(T : V \to W\) by \[ Tv = \lim _{i \to \infty } T_{i}v. \]

Show that \(T\) is bounded.

Raw LaTeX

Let \(V\), \(W\) be Banach spaces, and let \(T_{i} : V \to W\) be a sequence of bounded linear
operators such that \(\lim_{i \to \infty} T_{i}v\) exists for every \(v \in V\). Define the linear
map \(T : V \to W\) by
\[
Tv = \lim_{i \to \infty} T_{i}v.
\]


Show that \(T\) is bounded.

Topic Tags

Banach Spaces Bounded Linear Operators Lp Spaces and Sequence Spaces