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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2015 • Question 4

Question

Math 6210 • Real Analysis Spring 2015 • Question 4

If \(f:\mathbb R\to \mathbb C\) is in \(L^1(\lambda )\), show that the sequence of \(L^1(\lambda )\) functions \[ g_n(x)=f\left (x+\frac {1}{n}\right ) \] converges to \(f\) in \(L^1(\lambda )\).

Raw LaTeX

If \(f:\mathbb R\to\mathbb C\) is in \(L^1(\lambda)\), show that the
sequence of \(L^1(\lambda)\) functions
\[
g_n(x)=f\left(x+\frac{1}{n}\right)
\]
converges to \(f\) in \(L^1(\lambda)\).

Topic Tags

Lp Spaces and Sequence Spaces