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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2002 • Question 1

Question

Math 6210 • Real Analysis 2002 • Question 1

Let \(\alpha (x,y)\) be a continuous function on \([0,1]\times [0,1]\). Show that \[ A(f)(x)=\int _0^1 \alpha (x,y)f(y)\,dy \] defines a bounded operator from \(L^2([0,1])\) to \(C([0,1])\), considered as a Banach space with respect to \(\|f\|=\sup _x |f(x)|\).

Raw LaTeX

Let \(\alpha(x,y)\) be a continuous function on \([0,1]\times[0,1]\).
Show that
\[
A(f)(x)=\int_0^1 \alpha(x,y)f(y)\,dy
\]
defines a bounded operator from \(L^2([0,1])\) to \(C([0,1])\), considered as a
Banach space with respect to \(\|f\|=\sup_x |f(x)|\).

Topic Tags

Banach Spaces Lebesgue Integration Lp Spaces and Sequence Spaces