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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1993 • Question 4

Question

Math 6210 • Real Analysis Fall 1993 • Question 4

For \(f:\mathbb R\to \mathbb C\), and \(t\in \mathbb R\) define \[ f_t(x)=f(x-t). \] Suppose \(f\) is continuous with compact support. Prove directly that \(t\to f_t\) is a continuous map of \(\mathbb R\) into \(L^1(\mathbb R)\).

Raw LaTeX

For \(f:\mathbb R\to\mathbb C\), and \(t\in\mathbb R\) define
\[
f_t(x)=f(x-t).
\]
Suppose \(f\) is continuous with compact support. Prove directly that \(t\to f_t\)
is a continuous map of \(\mathbb R\) into \(L^1(\mathbb R)\).

Topic Tags

Compactness in Metric and Normed Spaces Lp Spaces and Sequence Spaces