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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2009 • Question 4

Question

Math 6210 • Real Analysis Fall 2009 • Question 4

Let \((X,\mu )\) be a finite measure space, and let \(f\in L^\infty (X)\). For \(g\in L^1(X)\), define \[ Tg=\int fg\,d\mu \] Show that \(T\) is a bounded linear functional on \(L^1(X)\), and that \(\lVert T\rVert =\lVert f\rVert _\infty \).

Raw LaTeX

Let \((X,\mu)\) be a finite measure space, and let
\(f\in L^\infty(X)\). For \(g\in L^1(X)\), define
\[
Tg=\int fg\,d\mu
\]
Show that \(T\) is a bounded linear functional on \(L^1(X)\), and that
\(\lVert T\rVert=\lVert f\rVert_\infty\).

Topic Tags

Bounded Linear Operators Dual Spaces and Hahn-Banach Theorem Lp Spaces and Sequence Spaces Measure Theory