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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2007 • Question 2

Question

Math 6210 • Real Analysis Fall 2007 • Question 2

Assume that \(\mathbb R\) has the Lebesgue measure. Let \(f\in L^\infty (\mathbb R)\). For \(g\in L^2(\mathbb R)\) define \[ M(g)=fg \] Prove that \(M:L^2(\mathbb R)\to L^2(\mathbb R)\) is a bounded linear operator, and that \(\lVert M\rVert =\lVert f\rVert _\infty \).

Raw LaTeX

Assume that \(\mathbb R\) has the Lebesgue measure. Let
\(f\in L^\infty(\mathbb R)\). For \(g\in L^2(\mathbb R)\) define
\[
M(g)=fg
\]
Prove that \(M:L^2(\mathbb R)\to L^2(\mathbb R)\) is a bounded linear operator,
and that \(\lVert M\rVert=\lVert f\rVert_\infty\).

Topic Tags

Bounded Linear Operators Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Maximum Modulus Principle