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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 3

Question

Math 6210 • Real Analysis Spring 2009 • Question 3

Let \(X\) and \(Y\) be Banach spaces, and suppose that \(T:X\to Y\) is a compact surjective linear map. Show that \(Y\) is finite dimensional. (Recall that a linear map is compact if the image of any bounded set has compact closure.)

Raw LaTeX

Let \(X\) and \(Y\) be Banach spaces, and suppose that
\(T:X\to Y\) is a compact surjective linear map. Show that \(Y\) is finite
dimensional. (Recall that a linear map is compact if the image of any bounded
set has compact closure.)

Topic Tags

Banach Spaces Bounded Linear Operators Compactness in Metric and Normed Spaces