Soft Question: I know a lot of people dislike the definition theorem proof style of books but I love them. I don't mean the diffuclty of the orignal Bourbaki books, but the fact that they are self-contained. So I am asking additionally for those types of books that don't omit details, steps or proofs for the theorems, books like those of Loring Tu, Munkres, Cohn, etc.
To be clear about which area I am looking for recommendations:
Algebraic topology, a book that covers the basics of the different types of fundamental groups, not just the first one, and invariance of domain
Lie groups(Hall is more matrix groups not necessarily the abstract groups one sees in differential geometry), like the proof of the existence and uniqueness of the exponential map. The covering group trick.
Differential Equations In the case of the differential equations does anyone knows of books that prove that the maximum number of linearly independent solutions is given by the dimension of the linear system of equations. I am looking for books like that but the ones that get recommended always focus on more practical things.
I am specially looking for ones that cover the concept of a weak solution, in the sense of measure theory
Differential Geometry: Distributions and the generalized theorem of calculus, and manifolds with corners. Hopf-Rinow theorem
Integration: Functional derivatives and the Riesz theorem, conditions for the existence of optima, like the Euler Lagrange Equations.
Schur's lemma for representation theory, not just the case for unitary matrices
I don't know if I should split up this questions since it is my first question. Thanks for any info and if it is against the rules I can split the question
