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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2003 • Question 3

Question

Math 6210 • Real Analysis 2003 • Question 3

Prove that if \(X\) is a Banach space with a closed unit ball which is compact, then \(X\) must be finite dimensional. Hint: consider the family of sets of the form \(\{x\in X:\|x\|=1,\text { and }f(x)=0\}\) where \(f\) is a bounded nonzero linear functional on \(X\).

Raw LaTeX

Prove that if \(X\) is a Banach space with a closed unit ball which is
compact, then \(X\) must be finite dimensional. Hint: consider the family of sets
of the form \(\{x\in X:\|x\|=1,\text{ and }f(x)=0\}\) where \(f\) is a bounded
nonzero linear functional on \(X\).

Topic Tags

Banach Spaces Bounded Linear Operators Compactness in Metric and Normed Spaces Zeros of Holomorphic Functions and the Identity Theorem