Real analysis: Real Analysis, Royden & Fitzpatrick|Principles of Real Analysis, Aliprantis and Burkinshaw
Differential Equation: Theory of Ordinary Differential Equations, Coddington and Levinson| Ordinary Differintial Equations Miller & Michel| Partial Differential Equations, Fritz John| Partial Differential Equations, Evans
Applied Math: Applied Analysis, Hunter
Functional Analysis: Functional Analysis, Peter D. Lax
Numerical Analysis: Numerical Analysis, Hammerlin
Finite Element: Mathematical Theory of Finite Element Methods, Brenner