graycat

W. Rudin's, as I recall, third edition, Principles of Mathematical Analysis.

Rudin, Principles of Mathematical Analysis, Third Edition.

the first half of Rudin's Real and Complex Analysis.

Rudin, Real and Complex Analysis

Rudin, 'Real and Complex Analysis'

For basic abstract algebra Herstein, Topics in Algebra should usually be enough and is well written.

Heavily about how what people do is in response to the anxiety they feel

Stephen Boyd and Lieven Vandenberghe, Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

and then Halmos (a finite dimensional introduction to Hilbert space

Halmos, Finite Dimensional Vector Spaces (FDVS).

Halmos, 'Finite Dimensional Vector Spaces'

Random variables and the associated probability theory and conditional probability theory are surprisingly relevant

Royden is breathtakingly gorgeous.

Fleming's Functions of Several Variables does it with both measure theory and exterior algebra.

I can recommend the beginning sections on vector spaces, convexity, and the inverse and implicit function theorems in Fleming, Functions of Several Variables

Royden's Real Analysis

Royden, Real Analysis sweetheart writing on that math.

his elementary An Introduction to Ordinary Differential Equations is nicely done

George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems