A Bittersweet Life

Kim Sun-woo (Lee Byung-hun) is a high ranking mobster and enforcer for Kang (Kim Yeong-cheol), a cold and calculating crime boss to whom he is unquestionably loyal. The two share concerns over business tensions with Baek Dae-sik (Hwang Jung-min), a son from a rival family, which is when Kang assigns Sun-woo what is perceived to be a simple errand while he is away on a business trip â to shadow his young mistress, Hee-soo (Shin Min-ah), whom he fears is having an "affair" with another man, giving Sun-woo the mandate to kill her (and her paramour) if he manages to discover it. As he performs his duty â following Hee-soo, and escorting her to a music recital one day â he becomes quietly enthralled by the girl's beauty and innocence as glimpses into his lonely, empty personal life become more prevalent. When he does come to discover Hee-soo's lover directly in her home, he fiercely beats him and prepares to inform Kang, but his attraction to her causes him to hesitate. He thus spares the two on the condition that they no longer see each other again, earning him Hee-soo's enmity.
Meanwhile, Sun-woo continues to be embroiled in personal business with Baek Dae-sik over having beaten up several of his henchmen earlier for overstaying their welcome at the hotel. He is then threatened by one of his enforcers to apologize, but he adamantly refuses, fueled by his frustrations over Hee-soo. As he relaxes in his apartment later one night, he is suddenly kidnapped by Baek's men to be tortured, but before they can do so they receive new orders via phone call and he is abruptly carried off to Kang, who has returned from overseas and has found out about his attempted cover-up of Hee-soo's affair. Kang's men torture him into confessing why he lied until he is left alone to think about his answer. A daring but messy escape follows, after which Sun-woo plans his revenge.
Help from one of Sun-woo's loyal men provides him with money and new clothes to go about his plan: he secretly delivers Hee-soo a gift to make amends and sets up a meeting with some local arms dealers, but as they are affiliated with Kang's organization he ends up killing them over a deal to buy a handgun â this incurs a vendetta with the brother of one of the dealers, who promptly sets out to find Sun-woo. He then goes on to set up a veiled rendezvous with Baek Jr. and kills him after a brief conversation, but he is viciously stabbed in the process. Bleeding, his violent shooting spree leads directly to Kang amidst one of his business meetings, where he vents bitterly over how badly he has been treated despite his seven years of service. Kang does not answer, and instead asks if Sun-woo's actions were directly because of Hee-soo. Sun-woo then shoots him, prompting a shootout with Baek Dae-sik's henchmen, who had quickly picked up his trail.
Sun-woo emerges as the only survivor of the battle with the arms dealer's brother finally catching up to him in the same room. Now dying from multiple gunshot wounds, he calls Hee-soo and pauses to reminisce on his only day with her, when he had escorted her to her music recital; in his memory, as he watches her play her cello, he finds himself overwhelmed with emotion and, in a rare moment of contentment, he smiles for the first time in the entire film. As he sheds a tear over this memory, the brother of the arms dealer executes him.
The film ends with a continuation of an earlier scene of Sun-woo looking out of a window at the city below him. After making sure he's alone, he begins to shadowbox his reflection in the glass, looking very happy.

Question: Does Sun-woo kill Hee-soo's lover?
Answer: No