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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1999 • Question 4

Question

Math 6210 • Real Analysis 1999 • Question 4

A bounded linear transformation \(A\) from a Banach space \(X\) to a Banach space \(Y\) is said to be ”compact” if the image under \(A\) of the unit ball in \(X\) has compact closure in \(Y\). Prove that if a compact linear transformation is also surjective, then \(Y\) is finite dimensional (you may assume that a locally compact Banach space must be finite dimensional).

Raw LaTeX

A bounded linear transformation \(A\) from a Banach space \(X\) to a
Banach space \(Y\) is said to be "compact" if the image under \(A\) of the unit ball
in \(X\) has compact closure in \(Y\). Prove that if a compact linear transformation
is also surjective, then \(Y\) is finite dimensional (you may assume that a
locally compact Banach space must be finite dimensional).

Topic Tags

Banach Spaces Bounded Linear Operators Compactness in Metric and Normed Spaces