star wars - Who/what exactly does Darth Vader believe taught Luke between the events of “The Empire Strikes Back” and “Return of the Jedi?”

Re-watched Return of the Jedi (1983 non-“Special” version). Near the beginning of the duel between Luke and Darth Vader on Death Star II, just after Luke kicks Darth Vader down some stairs, Darth Vader recomposes himself and says the following after Luke turns off his own lightsaber; video clip below:

Obi-Wan has taught you well.

Okay, Luke was clearly taught by Yoda during the events of The Empire Strikes Back. The audience knows that and only Luke, Obi-Wan and R2-D2 seem to share that knowledge of Yoda; past that utterly nobody else seems to know about Yoda. So it’s pretty clear that Darth Vader had 100% no knowledge of who Yoda was/is or even if Yoda is still alive. But still, even with Yoda’s guidance Luke was clearly underprepared and overpowered in The Empire Strikes Back.

So given the fact that Darth Vader clearly knows that he killed Ben (Obi-Wan) Kenobi during the first Star Wars film and he also clearly knows how raw and rough Luke’s skills were during The Empire Strikes Back, how exactly does Darth Vader believe Luke has been “taught” by the time the events of Return of the Jedi happen?

Luke is clearly better prepared for a serious Jedi versus Sith confrontation in Return of the Jedi. So does Darth Vader assume something along the lines of Obi-Wan’s “Force Ghost” has somehow come back to guide Luke on his journey to becoming a Jedi?

Or is Darth Vader still somehow obsessed with Ben (Obi-Wan) Kenobi being someone who he was in personal conflict with—as shown in the prequels—as well as someone who clearly exposed naive farm boy Luke Skywalker to the world of the Jedi? Meaning without Obi-Wan’s assistance/interference/meddling, Luke would just be some dumb-hick farm boy who was a threat to nobody in the middle of nowhere.

Answer

From James Kahn's novelization:

NOTE THAT

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  THIS HAPPENS BEFORE THE FIGHT. Right after Luke entered his presence. Which means before Vader finds out Luke's skills improved as per OP's curiosity

  
  


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  Obviously, Vader was standing right by him and heard what the Emperor said. So, he knew that Luke was trained by Yoda before he found out how much Luke improved

  
  

The moment passed. He did nothing.

“Tell me, young Skywalker,” the Emperor said when he saw Luke’s first struggle had taken its course. “Who has been involved in your training until now?” The smile was thin, open-mouthed, hollow.

Luke was silent. He would reveal nothing.

“Oh, I know it was Obi-Wan Kenobi at first,” the wicked ruler continued, rubbing his fingers together as if trying to remember. Then pausing, his lips creased into a sneer. “Of course, we are familiar with the talent Obi-Wan Kenobi had, when it came to training Jedi.” He nodded politely in Vader’s direction, indicating Obi-Wan’s previous star pupil. Vader stood without responding, without moving.

Luke tensed with fury at the Emperor’s defamation of Ben—though, of course, to the Emperor it was praise. And he bridled even more, knowing the Emperor was so nearly right. He tried to bring his anger under control, though, for it seemed to please the malevolent dictator greatly.

Palpatine noted the emotions on Luke’s face and chuckled. “So, in your early training you have followed your father’s path, it would seem. But alas, Obi-Wan is now dead, I believe; his elder student, here, saw to that—” again, he made a hand motion toward Vader. “So tell me, young Skywalker—who continued your training?”

That smile, again, like a knife. Luke held silent, struggling to regain his composure.

The Emperor tapped his fingers on the arm of the throne, recalling. “There was one called … Yoda. An aged Master Jed … Ah, I see by your countenance I have hit a chord, a resonant chord indeed. Yoda, then.”

Luke flashed with anger at himself, now, to have revealed so much, unwillingly, unwittingly. Anger and self-doubt. He strove to calm himself—to see all, to show nothing; only to be.

“This Yoda,” the Emperor mused. “Lives he still?”

Luke focused on the emptiness of space beyond the window behind the Emperor’s chair. The deep void, where nothing was. Nothing. He filled his mind with this black nothing. Opaque, save for the occasional flickering of starlight that filtered through the ether.

“Ah,” cried Emperor Palpatine. “He lives not Very good, young Skywalker, you almost hid this from me. But you could not. And you can not. Your deepest flickerings are to me apparent. Your nakedest soul. That is my first lesson to you.” He beamed.

Please note that the novelization (which is fully Disney canon) does NOT have the famous movie line "Obi-Wan has taught you well". In the place where it is in the film, we just have:

The Emperor, watching joyously, saw this, and goaded Luke on to revel in his Darkness. “Use your aggressive feelings, boy! Yes! Let the hate flow through you! Become one with it, let it nourish you!”

Luke faltered a moment—then realized what was happening. He was suddenly confused again. What did he want? What should he do? His brief exultation, his microsecond of dark clarity—gone, now, in a wash of indecision, veiled enigma. Cold awakening from a passionate flirtation.

He took a step back, lowered his sword, relaxed, and tried to drive the hatred from his being.

In that instant, Vader attacked. He lunged half up the stairs, forcing Luke to reverse defensively. He bound the boy’s blade with his own, but Luke disengaged and leaped to the safety of an overhead gantry. Vader jumped over the railing to the floor beneath the platform on which Luke stood.

“I will not fight you, Father,” Luke stated.

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For completeness, Junior Novelization by Ryder Windham does NOT have Kahn's story progression, and instead faithfully follows the film's dialog (it basically is pretty much a movie script's copy), with this training mind-reading by the Emperor NOT happening, and with the famous "Obi-Wan has taught you well" line being the only clue as to Vader's thoughts.

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Having said that, it is clear why Obi-Wan's training is what Vader cares about, and not someone else's.

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  He was Obi-Wan's prior student. Now, Luke's taken his place. It's a rivalry.

  
  


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  If we include Episode III canon, he has a really really intense relationship with Obi-Wan. Brother. Teacher. Teacher he loves to hate. In the very end, someone Anakin is even jealous of as far as Padme, when they both get out of Padme's ship. Obi-Wan is the center of Anakin's world, and Yoda (or anyone else) is not anywhere close in his concerns.