Could lecturing from a textbook constitute copyright infringement?
I can't imagine an existing legal regime in which teaching from a textbook and &quot;publishing&quot; the lectures would be considered wrong unless the instructor explicitly made images or snippets copied from the book open to view. Likewise reading from the text verbatim would possibly be wrong depending on how much was read (fair use principles are discussed below). For a sentence or two, there are likely no issues. For a section, beware.
Referring the students to certain page/section numbers in a text is completely innocuous. Putting up page images is problematic.
Paging through a book with the camera pointed at the pages would probably be infringement everywhere, unless the snippets were very small (fair use). But, such paging seems to be blatant misuse in any case. Note, also, that the fair use rules for educational purposes are (normally) a bit relaxed from the general rules, though this has been changing.
But the lecture &quot;belongs&quot; to the lecturer, not the author/publisher of the book.
In some fields the standard for claiming a violation is much higher. If I copy the definition of the derivative (Calculus) onto a whiteboard, I'm not violating copyright even if my lecture is widely available. But that is because of a (typical) exception that things that can be said only one (or a very few) ways can't be copyrighted in the first place. So the author of Rudin (i.e. Rudin) used this exception in writing the book in the first place. But if I copy an equivalent amount from, say, a philosophy or history book, then there might be a problem.
You are much more likely to get a takedown notice at YouTube if you have some music playing in the background as you lecture. But that is because of draconian laws put in place by media publishers, not just copyright.

One of the (typical) standards for fair use is whether the use reduces the market for the item copied. If it does, then there are issues, otherwise there are not likely to be. If you &quot;teach from&quot; the book in such a way that students don't need access to it, then you should reexamine what you are doing. But even that implies that the &quot;teaching from&quot; is extensive. For supplementary materials, it might not be necessary for students to have other access. But there would be a &quot;quantity&quot; assessment made by anyone wanting to claim infringement and any court judging it. Does your use reduce the market?
And note that the laws are different everywhere, hence my use of &quot;typical&quot; etc. in a few places here. Perhaps a king could copyright the alphabet and make all written text subject to rules. See the book Fahrenheit 451 by Ray Bradbury for an extreme example.

I'll also note that, done right, such a course could actually increase the market for a textbook.