caesium

I used this for my undergrad analysis class and then relied on it as a foundation for reading Rudin/Royden in grad school. Killed qualifying exams. Use it. Sure, it's all Riemann integration but you will not suffer until you get to measure-theoretic integration later.

Rosenlicht. Linear Algbera done Right. Then Finite Dimensional Vector Spaces. Halmos' Measure Theory. Awesome.