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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2025 • Question 5

Question

Math 6210 • Real Analysis Spring 2025 • Question 5

Let \(V,W\) be Banach spaces, and let \(T_i:V\to W\) be a sequence of continuous 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 continuous.

Raw LaTeX

Let \(V,W\) be Banach spaces, and let \(T_i:V\to W\) be a sequence of
continuous 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 continuous.

Topic Tags

Banach Spaces Bounded Linear Operators Lp Spaces and Sequence Spaces