In (the) notes he gave me, I learned that

In (the) notes he gave me, I learned that
 

-In (the) notes he gave me, I learned that in the Qing dynasty, province governors were appointed by the emperor himself.

I'd use THE, but is there any instance where we can omit THE?

Using the definite article means that the speaker is referring to specific notes already mentioned or known.

It's possible to drop the definite article when referring to notes in general.
In notes taken at the time of the accident, ...
From notes found in her journal, ...
In notes from around the globe, ...

In notes he gave me, ... (not any notes in particular, but notes in general)
In the notes he gave me, ... (a specific set of notes, as opposed to other notes he may have written)

But, for example, for NOTES FOUND IN HER JOURNAL, isn't the thing being referred to very definite?

Not necessarily. :thinking:
"The notes found in her journal" could refer to all of them, while "notes found in her journal" may talk about only a few of those notes.

Pardon my quoting a long paragraph, because without doing so, the usage THE is difficult to discuss:

-We had all been warned to appear before the magistrates upon the Thursday; but when the Thursday came there was no occasion for our testimony. A higher Judge had taken the matter in hand, and Jefferson Hope had been summoned before a tribunal where strict justice would be meted out to him. On the very night after his capture the aneurism burst, and he was found in the morning stretched upon the floor of the cell, with a placid smile upon his face, as though he had been able in his dying moments to look back upon a useful life, and on work well done.


According to Rusty, am I correct in saying ON WORK WELL DONE refers to his work generally, while ON THE WORK WELL DONE would refer to, or be as opposed to, some particular work HE had done?