Linear Algebra. Complex numbers. Differential and integral calculus of functions of real variables.
Basic functional analysis: normed and Banach spaces, Hilbert spaces, Lebesgue integral, linear operators, weak topologies, distribution theory, Sobolev spaces, fixed point theorems, calculus in Banach spaces, spectral theory.
Applications: ordinary differential equations, boundary value problems for partial differential equations, linear system theory, optimization theory.
Ruth F. Curtain, A.J. Pritchard, Functional Analysis in Modern Applied Mathematics, Academic Press, 1977