Hilbert-Schmidt integral operator is compact.

In what follows, we will prove that every Hilbert-Schmidt integral operator is compact.
Formally, let $k\in L_2([0,1]\times[0,1])$ and $T:L_2[0,1]\to L_2[0,1]$ be defined by
$$Tf(x):=\int_0^1k(x,y)f(y)dy.$$
Then $T$ is compact operator.

Although the proof may seem a bit long at first sight, it is proceeds naturally, utilizing approximation of $k$ with continuous functions.

$\textbf{Proof}$. We approximate $k$ with a sequence of continuous functions, then prove that each of the corresponding operators has image contained in $C[0,1]$, then show that each of these operators is compact when $C[0,1]$ is equipped with the $\sup$ norm. This will be done via the Arzela-Ascoli theorem. Finally show that these converge uniformly (in the operator norm) to the original operator, and since they are compact, the original must be as well.
The Cauchy-Bunyakovsky-Schwartz inequality is used throughout the proof.

Thus we start by choosing a sequence $(k_n)_{n\ge 1}$ of continuous functions converging to $k$ in the sense of $L_2$ norm, i.e.
$$\lim_{n\to\infty}\int_0^1(k_n(x)-k(x))^2\text{d}x=0.$$
Such a sequence could be constructed using the Fourier decomposition of $k$ into sines for example.
Then for any $n\ge 1$ define $$T_nf(x):=\int_0^1k_n(x,y)f(y)dy.$$
Fix $m\ge 1$ and for brevity denote $S=T_m$.
First we show that $\text{Im} S\subseteq C[0,1]$. Let $f\in L_2[0,1]$. We aim to show that $Sf$ is a continuous function. Indeed, let $x\in [0,1]$ and $(x_n)_{n\ge 1}$ be a sequence in $[0,1]$ converging to $x$. Then, if we denote $g_n(y)=k_m(x_n,y)$, we have
$$\lim_{n\to\infty}g_n(y)=k_m(x,y)$$
Moreover, if $\displaystyle C:=\sup_{x\in[0,1]}|k_m(x,y)|$ (finite, since $k_m$ is continuous), then $|g_n(y)|=|k_m(x_n,y)|\le C$. Thus Dominated convergence theorem is applicable, hence
$$\lim_{n\to\infty}Sf(x_n)=\lim_{n\to\infty}\int_0^1g_n(y)f(y)\text{d}y=\int_0^1k_m(x,y)f(y)\text{d}y=Sf(x).$$
So $S:L_2[0,1]\to C[0,1]$.
Next, equip $C[0,1]$ with the $\sup$ norm. We will prove $S$ is compact. To this end, we take a sequence $(f_n)_{n\ge 1}\subset L_2[0,1]$, such that $\|f_n\|_2\le 1$ for all $n$, and we are going to prove that $(Sf_n)_{n\ge 1}$ has a convergent subsequence in $C[0,1]$. The criterion telling us whether this is true is the Arzela-Ascoli theorem.
From the Cauchy-Bunyakovsky-Schwartz inequality we obtain $$|Sf_n(x)|=\left|\int_0^1 k_m(x,y)f_n(y)\text{d}y\right|\le \sqrt{\int_0^1k_m(x,y)^2\text{d}y\int_0^1f_n(y)^2\text{d}y}\le C$$
(for $\displaystyle C:=\sup_{x\in[0,1]}|k_m(x,y)|$) and thus the sequence $(Sf_n)_{n\ge 1}$ is uniformly bounded.
Moreover, let $\varepsilon>0$. Then there is some $\delta>0$ such that whenever $|x_1-x_2|< \delta$, then $|k_m(x_1,y)-k_m(x_2,y)|<\varepsilon$ for all $y\in[0,1]$ (uniform continuity of continuous functions).Thus for any $n$ and any $x_1,x_2$, such that $|x_1-x_2|<\delta$ we have
$$|Sf_n(x_1)-Sf_n(x_2)|=\left|\int_0^1(k_m(x_1,y)-k_m(x_2,y))f_n(y)\text{d}y\right|$$
$$\le \sqrt{\int_0^1(k_m(x_1,y)-k_m(x_2,y))^2\text{d}y \int_0^1 f_n(y)^2\text{d}y}\le \varepsilon$$
This shows that the sequence $(Sf_n)_{n\ge 1}$ is equicontinuous. Thus it contains convergent subsequence.
This concludes the proof that $(T_m)_{m\ge 1}$ is a sequence of compact operators (acting from $L_2[0,1]$ to $C[0,1]$). Moreover, for any $f\in C[0,1]$, $\|f\|_{\infty}\ge \|f\|_2$, which means that the $L_2$ topology on $C[0,1]$ is weaker than the uniform topology. Hence the operators $T_m$, $m\ge 1$, are compact, even if we consider them from $L_2[0,1]$ to $L_2[0,1]$.
It remains to observe that $T_m$ converges to $T$ in the operator norm.
Let $f\in L_2[0,1]$. Then
$$\|T_mf-Tf\|_2^2=\int_0^1\left(\int_0^1(k(x,y)-k_m(x,y))f(y)\text{d}y\right)^2\text{d}x\le$$
$$\int_0^1\int_0^1(k(x,y)-k_m(x,y))^2\text{d}y\int_0^1f(y)^2\text{d}y\text{d}x=\int_0^1\int_0^1(k(x,y)-k_m(x,y))^2\text{d}y\text{d}x\int_0^1f(y)^2\text{d}y=$$
$$\|k_m-k\|_2^2\|f\|_2^2.$$
This shows that
$$\|T_m-T\|\le \|k_m-k\|_2,$$
and since $(k_m)_{m\ge 1}$ was taken to converge to $k$ in $L_2$, then $T_m$ converges to $T$.
The proof is finished, by saying, that the space of compact operators is closed, and as $T_m$ are compact, so is $T$.

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