I'm having some issues understanding a proof that a Hilbert-Schmidt integral operator is bounded. I'll start off by giving the definitions I'm working with.
Definition of integral operator:
$\textit{Let k be a function of two variables (x,t)}\in I\times I=I^2\textit{ where I is a finite or infinite real interval. We define a linear integral operator K with kernel k(x,y) as}$
$$Ku(x)=\int_{I}k(x,y)u(y)\ dy,\ x\in I$$
Definition of Hilbert-Schmidt integral operator:
$\textit{An integral operator on }L^{2}(I)\textit{ is called a Hilbert-Schmidt operator if the kernel k is in }L^2(I\times I)\textit{,}$
$\textit{that is if}$
$$||k||^2=\int_{I}\int_{I}|k(x,y)|^2\ dxdy<\infty$$
The book now proceeds to show that a Hilbert-Schmidt integral operator is bounded with $||K||\leq||k||$. In the proof the following is stated:
$\textit{Assume that K is a Hilbert-Schmidt operator with kernel k}\in L^2(I\times I). \textit{Let u}\in L^2(I),$
$\textit{then from Fubini's theorem in measure theory, the function }x\mapsto \int_{I}|k(x,y)|^2\ dy$
$\textit{is in }L^1(I)\textit{ such that }\int_{I} was \int_{I}|k(x,y)|^2\ dxdy<\infty.\textit{ Thus it is allowed to use the Cauchy-Schwarz inequality such that}$
$$|Ku(x)|=|\int_{I}k(x,y)u(y)\ dy|\leq ||k(x,\cdot)||\ ||u||$$
$\textit{since }||\overline{u}||=||u||.$
I' having some difficulties understanding this argument. The remaining part of the proof is not that difficult once this argument is clear. Can anyone give some hints as to why this makes sense?
Thanks in advance!
Best Answer
You don't say what exactly are your difficulties, so I will proceed step by ste. By the Cauchy-Schwarz inequality\begin{align} |Ku(x)|&=\Bigl|\int_{I}k(x,y)\,u(y)\,dy\Bigr|\\ &\le\int_{I}|k(x,y)|\,|u(y)|\,dy\\ &\le\Bigl(\int_{I}|k(x,y)|^2\,dy\Bigr)^{1/2}\Bigl(\int_{I}|u(y)|\,dy\Bigr)^{1/2}\\ &=\Bigl(\int_{I}|k(x,y)|^2\,dy\Bigr)^{1/2}\|u\|. \end{align} Then $$ |Ku(x)|^2\le\Bigl(\int_{I}|k(x,y)|^2\,dy\Bigr)\|u\|^2 $$ and $$ \|Ku\|^2=\int_I|Ku(x)|^2dx\le\Bigl(\int_I\Bigl(\int_{I}|k(x,y)|^2\,dy\Bigr)dx\Bigr)\|u\|^2=\|k\|^2\|u\|^2, $$ that is, $$\|Ku\|\le\|k\|\,\|u\|.$$