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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 4

Question

Math 6210 • Real Analysis Spring 2020 • Question 4

Let \(C([0,1])\) be the Banach space of continuous functions on \([0,1]\) with the sup norm and view it as a natural subspace of \(L^\infty ([0,1])\), where \([0,1]\) is given Lebesgue measure, and the norm is essential supremum. Prove the existence of a bounded functional on \(L^\infty ([0,1])\) which is not identically \(0\) but vanishes on \(C([0,1])\).

Raw LaTeX

Let \(C([0,1])\) be the Banach space of continuous functions on
\([0,1]\) with the sup norm and view it as a natural subspace of
\(L^\infty([0,1])\), where \([0,1]\) is given Lebesgue measure, and the norm is
essential supremum. Prove the existence of a bounded functional on
\(L^\infty([0,1])\) which is not identically \(0\) but vanishes on \(C([0,1])\).

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Topic Tags

Banach Spaces Isolated Singularities and Residues Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces