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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 2

Question

Math 6210 • Real Analysis Spring 2008 • Question 2

Let \(K\) be a continuous function on the square \([0,1]\times [0,1]\), and let \([0,1]\) be endowed with the Lebesgue measure. For \(f\in L^2([0,1])\) define \[ Tf(x)=\int _0^1 K(x,y)f(y)\,dy. \] Show that \(Tf\in L^2([0,1])\) and that \(T:L^2\to L^2\) is a bounded operator with \(\lVert T\rVert \le \lVert K\rVert _\infty \).

Raw LaTeX

Let \(K\) be a continuous function on the square \([0,1]\times[0,1]\),
and let \([0,1]\) be endowed with the Lebesgue measure. For
\(f\in L^2([0,1])\) define
\[
Tf(x)=\int_0^1 K(x,y)f(y)\,dy.
\]
Show that \(Tf\in L^2([0,1])\) and that \(T:L^2\to L^2\) is a bounded operator
with \(\lVert T\rVert\le\lVert K\rVert_\infty\).

Topic Tags

Lebesgue Integration Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory