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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2009 • Question 5

Question

Math 6210 • Real Analysis Fall 2009 • Question 5

Either prove the following statement if true, or prove that it is false by providing a counterexample. Let \(\{f_n\}\) be a sequence in \(L^2([0,1])\) such that \(\lVert f_n\rVert _2\to 0\). Then \(f_n\to 0\) almost everywhere.

Raw LaTeX

Either prove the following statement if true, or prove that it is
false by providing a counterexample. Let \(\{f_n\}\) be a sequence in
\(L^2([0,1])\) such that \(\lVert f_n\rVert_2\to0\). Then \(f_n\to0\) almost
everywhere.

Topic Tags

Almost Everywhere and Convergence in Measure Lp Spaces and Sequence Spaces