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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2024 • Question 4

Question

Math 6210 • Real Analysis Spring 2024 • Question 4

Prove that if \(A:\ell ^2(\mathbb {N})\to \ell ^2(\mathbb {N})\) is continuous and linear and \(AB(0,1)\) is dense in \(B(0,1)\) then \(A\) is surjective. Note that this is true for Banach spaces as well (though we are not asking you to prove it).

Raw LaTeX

Prove that if \(A:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N})\) is continuous and linear and \(AB(0,1)\) is
dense in \(B(0,1)\) then \(A\) is surjective. Note that this is true for Banach
spaces as well (though we are not asking you to prove it).

Topic Tags

Banach Spaces Compactness in Metric and Normed Spaces Lp Spaces and Sequence Spaces