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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2008 • Question 5

Question

Math 6210 • Real Analysis Fall 2008 • Question 5

Let \(T:C_0(\mathbb R)\to C_0(\mathbb R)\) be the translation operator defined by \((T.f)(x)=f(x+1)\). Show that there is no non-zero bounded linear functional \(\Phi :C_0(\mathbb R)\to \mathbb C\) invariant under \(T\), i.e. such that \(\Phi \) satisfies \[ \Phi (f)=\Phi (Tf). \]

Raw LaTeX

Let \(T:C_0(\mathbb R)\to C_0(\mathbb R)\) be the translation
operator defined by \((T.f)(x)=f(x+1)\). Show that there is no non-zero bounded
linear functional \(\Phi:C_0(\mathbb R)\to\mathbb C\) invariant under \(T\), i.e.\
such that \(\Phi\) satisfies
\[
\Phi(f)=\Phi(Tf).
\]

Topic Tags

Bounded Linear Operators Dual Spaces and Hahn-Banach Theorem Lebesgue Measure and Translation Invariance Zeros of Holomorphic Functions and the Identity Theorem