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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2016 • Question 3

Question

Math 6210 • Real Analysis Fall 2016 • Question 3

Consider the interval \([-1,1]\) with a finite Borel measure \(\mu \) (not assumed to be the Lebesgue measure). Let \(C([-1,1])\) be the space of continuous functions endowed with the \(L^2\) norm, and let \(T:C([-1,1])\to \mathbb R\) be the operator \(T(f)=f(0)\). Show that \(T\) is a bounded operator (with respect to the \(L^2\) norm) if and only if \(\mu (\{0\})>0\).

Raw LaTeX

Consider the interval \([-1,1]\) with a finite Borel measure \(\mu\)
(not assumed to be the Lebesgue measure). Let \(C([-1,1])\) be the space of
continuous functions endowed with the \(L^2\) norm, and let
\(T:C([-1,1])\to\mathbb R\) be the operator \(T(f)=f(0)\). Show that \(T\) is a
bounded operator (with respect to the \(L^2\) norm) if and only if
\(\mu(\{0\})>0\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory