In principle the pensum is everything in Tom Lindstr?m's kompendium ''Mathematical Analysis'', except the following:
-- Section 3.6 about the alternative characterization of compactness
-- Section 3.7 about completion of metric spaces.
-- Section 4.8 about the Arzela-Ascoli theorem.
-- Sections 5.6 and 5.7 about Baire's category theorem and its consequences.
-- Sections 6.7 to 6.11 about calculus in Normed Spaces.
But there might be exercises where you are guided through similar results...
I will probably have a slightly different approach to the following topics:
-- Sections 4.3 and 6.5 about integration.
-- Sections 4.10 and 4.11 about the Stone Weierstrass theorem.
-- Sections 4.7, 4.9 and 6.8 about differential equations.
Everything in my posted lecture notes is also pensum...
The course prolongates earlier Calculus courses, and such things are also part of the pensum. For instance everything in Tom Lindstr?m's book "Kalkulus" is pensum. The relevant material is covered in Stephen Abbott's book 'Understanding Analysis" (which is available for free on Springer-Link). I will use and refer to the latter during the semester.