Consider the Hilbert Schmidt operator $K: L^2(\Omega) \rightarrow L^2(\Omega)$, $\Omega \subset \subset \mathbb R^N$, with $k \in L^2(\Omega \times \Omega)$ and $f \in L^2(\Omega)$,
$$(Kf)(x) := \int_\Omega k(x,y)f(y)\, dy.$$
I want to show that the Hilbert Schmidt operator $K$ is a compact operator. Therefore I'm using this characterization.
Let $X$, $Y$ be normed linear spaces and $X$ reflexive. A continuous linear operator $T: X \rightarrow Y$ that maps weakly convergent sequences onto strongly convergent sequences is compact.
(We already know that $K$ is well-defined as is proven here.)
My question here is, isn't it obvious that $K$ is compact?
- We know that $K$ is linear and bounded, hence continuous.
- Every continuous map takes weakly convergent sequences to weakly convergent sequences.
- The norm itself is also continuous.
- Weak convergence together with convergence of the norms implies convergence.
Thus $K$ is compact. Am I missing something here? Or better: What am I missing here?
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I'm also adding the proof from the textbook for completeness:
Proof. Let $(f_n)_{n \in \mathbb N} \subset L^2(\Omega)$ a weakly convergent sequence, then $(f_n)_{n \in \mathbb N}$ is bounded. That is, $\exists C > 0 $ such that $||f_n||_{L^2(\Omega)} \leq C$, $\forall n \in \mathbb N$. By Fubini's theorem we have for almost every $x\in \Omega$ that $$ || k(x,\cdot) ||_{L^2(\Omega)} = \int_\Omega |k(x,y)|^2 \, dy < \infty .$$
Thus for almost every $x \in \Omega$ we have
$\begin{align}
\lim_{n \rightarrow \infty} (Kf_n)(x) & = \int_\Omega k(x,y)f_n(y) \, dy = \lim_{n \rightarrow \infty} \langle k(x,\cdot), f_n \rangle_{L^2(\Omega)} \\
& = \langle k(x,\cdot), f \rangle_{L^2(\Omega)} = \int_\Omega k(x,y)f(y) \, dy = (Kf)(x)
\end{align}$
By Cauchy-Schwarz's inequality we have $$ (Kf_n)(x) \leq ||f_n||_{L^2(\Omega)} \int_\Omega |k(x,y)|^2 \, dy \leq C \, \int_\Omega |k(x,y)|^2 \, dy $$
Hence by Lebesgue's dominant convergence theorem we have convergence of the norms $$ \lim_{n \rightarrow \infty} \int_\Omega |(Kf_n)(x)| \, dx = \int_\Omega |(Kf)(x)| \, dx ,$$
that is $|| Kf_n ||_{L^2(\Omega)} \rightarrow || Kf ||_{L^2(\Omega)}\, \, (n\rightarrow \infty)$. Since weak convergence together with (strong or normal) convergence of the norms implies (strong) convergence, $K$ is compact.
Best Answer
The norm is continuous as a map $\|\cdot\|: (X,\|\cdot\|_X)\to \mathbb R$ but not when defined on $X$ with its weak topology. This is where your general argumentation fails. The proof from your textbook is fine, however one can in general show that every Hilbert-Schmidt operator is already compact: One can represent the finite rank operators in a Hilbert space as a tensor product. We have several natural norms on this space, whose completions lead to several classes of operators (nuclear operators, Hilbert-Schmidt operators and compact operators) and those norms dominate each other in such a way that we have the inclusions nuclear operator is Hilbert-Schmidt operator is compact operator.