An operator is compact iff it maps any weakly convergent sequence to a convergent one. ﻿This was the proof from a textbook "Hilbert Space Operators in Quantum Physics" by Jiří Blank, Pavel Exner, Miloslav Havlíček.
3.5.1 Theorem: An operator is compact iff it maps any weakly convergent sequence to a convergent one.
Proof: Suppose that $C\in K(H)$ and $ x_n \to^{w} x$  does not imply $Cx_n → Cx.$ Then there is a positive and a growing sequence $\{n_k\}$ of natural numbers such that $||Cx - y_k||\ge ε$ holds for $y_k = Cx_{n_k}, k = 1, 2,....$ $\{x_{n_k}\}$ is bounded, so one can select from $\{y_k\}$ a subsequence $\{y_{k_j}\}$ which converges to some $y.$ This means, in particular, $y_{k_j} \to^{w} y$ . On the other hand $Cx_n\to^{w} Cx$  since $C$  is bounded, so together we get $y = Cx$ in contradiction with the assumption. The opposite implication follows from the reflexivity of H  one can select a weakly convergent subsequence $\{x_{n_k}\}$ from any bounded $\{x_n\}\subset H$, and $\{Cx_{n_k}\}$ converges by assumption.
My attempt of understanding the proof:-
We are trying to prove using the method of contradiction.
Suppose that $C\in K(H)$(This means $C$ is a compact operator.) and $ x_n \to^{w} x$  does not imply $Cx_n → Cx.$
We know that $Cx_n$does not converge to $Cx.$ Hence there exists a subsequence  of $x_n$ such that there is a positive and a growing sequence $\{n_k\}$ of natural numbers such that $||Cx - y_k||\ge ε$ holds for $y_k = Cx_{n_k}, k = 1, 2,....$
(Am I correct?)
Using the logic given below:-
We know that $x_n\to^w x \implies \{x_n\}$ is a bounded sequence.
Hence $\overline{\{Cx_n\}}$ is a compact set.(Because $C$ is a compact set.)
Hence, $\{Cx_n\}$ is bounded. $\{Cx_n\}$ is a sequence in Hilbert space. Hence, it is a sequence in complete metric space. Hence, Bounded sequence has convergent subsequence.
$\{x_{n_k}\}$ is bounded, so one can select from $\{y_k\}$ a subsequence $\{y_{k_j}\}$ which converges to some $y.$
Does convergence implies weakly convergence? why do we use weakly convergence in the range space? We kneed to prove $Cx_n$ converges to $Cx$ strongly. Right?
I am not able to understand the proof. Could you help me to understand the proof better?
 A: Convergence always implies weak convergence . If $x_{n}\to x$ then $f(x_{n})\to f(x)$ for any continuous linear functional just by continuity.

"Hence, it is a sequence in complete metric space. Hence, Bounded
sequence has convergent subsequence. "

This is wrong. This is true for finite dimensional normed spaces and is called the Bolzano Weiresstrass theorem. You can just check this yourself as it would imply that any sequence in the unit ball has a convergent subsequence and thus the unit ball would be compact which is not possible. The unit ball is compact if and only if the normed space is finite dimensional. Many proofs are available of this fact . You can try it yourself using Riesz Lemma.
Here is the complete proof of the Theorem
Let $x_{n}$ be an arbitrary sequence such that $||x_{n}||\leq 1$ . So $C(x_{n})$ is an arbitrary sequence in $C(B_{H})$ (the image of the unit ball) .
Using your very own question here you have that a bounded sequence in a Hilbert Space has a weakly convergent subsequence. Let $x_{n_{k}}$ be a weakly convergent subsequence  of $x_{n}$ .
Then as $C$ maps weakly convergent sequences to convergent ones, we have $C(x_{n_{k}})$ is convergent. Thus any sequence in $C(B_{H})$ has a convergent subsequence and thus $C(B_{H})$ is sequentially compact . Thus $C$ is a compact operator.
Conversely let $C$ be a compact operator. Let $x_{n}\rightharpoonup x$ . Then $x_{n}$ is bounded by Uniform Boundedness Principle and as $C$ is bounded, $C(x_{n})$ is bounded. Let $C(x_{n_{k}})$ be an arbitrary subsequence of $C(x_{n})$ . Then by compactness of $C$, there exists a further subsequence $C(x_{n_{k_{l}}})$ that converges to some $z$ say.
But for any continuous linear functional $g\in Y^{*}$ you have $g\circ C\in X^{*}$ . Thus $g\circ C(x_{n})\to g\circ C(x)$ . Thus $C(x_{n})\rightharpoonup C(x)$. Thus as convergence implies weak convergence, you have $C(x_{n_{k_{l}}})\rightharpoonup z$ . But by uniqueness of weak limit, you have $z=C(x)$ and hence $C(x_{n_{k_{l}}})\to z= C(x)$
Here we can use a basic result from real analysis that a sequence $x_{n}$ converges to $x$ in a metric space if and only if given any subsequence $x_{n_{k}}$ of $x_{n}$ there exists a further subseuqnece $x_{n_{k_{l}}}$ such that $x_{n_{k_{l}}}\to x$. Thus using this result we get $C(x_{n})\to C(x)$ .
The proof of the above is exactly the fact that is being used in your proof. Suppose $x_{n}$ does not converge to $x$. Then there exists a subsequence $x_{n_{k}}$ and a fixed $\epsilon >0$ such that $d(x_{n_{k}},x)\geq \epsilon\,,\forall k\in\Bbb{N}$ . But  $x_{n_{k}}$ has a subsequence $x_{n_{k_{l}}}\to x$ . Thus there exists $L$ such that $l\geq L\implies d(x_{n_{k_{l}}},x)<\frac{\epsilon}{2}$ which is a contradiction. Thus $x_{n}\to x$. Conversely if $x_{n}\to x$ then for any subseuquence $x_{n_{k}}\to x$.
