Question: Given my background, what are some of the possible second steps in studying functional analysis? I've been looking at the books listed here, but I can't figure out which books are good for a second and not a first course. If there are multiple second options to choose from, I think that I'd prefer those closer to statistics, numerical analysis, the calculus of variations, optimization more generally, and ODEs/PDEs.
My Background: I've worked through a fair portion of Royden (4th edition). I've worked through most of chapters 1 through 16 (Topics included, but not limited to: Lebesgue interaction, $L^{p}$ spaces, duality in $L^{p}$, metric and topological spaces, Linear operators, Duality, compactness, and convexity in normed spaces (Hahn-Banach, Mazur's Theorem, Krein-Milman), weak compactness, metriziability of weak topologies, Hilbert spaces, the spectral theorem in Hilbert spaces for compact self adjoint operators, Riesz-Schauder).
I've also studied all of baby Rudin, an ODEs course (Differential Equations, Dynamical Systems, and an Introduction to Chaos), most of the first 3 chapters of Evans, a finite element method course (including an intro to Sobolev spaces and their applications to elliptic PDEs), a year of algebra out of Dummit and Foote, the basics of complex analysis, and some numerical analysis.
Interests: Numerical PDE, optimization, and also in the interlocking history of functional analysis and statistics (e.g. through using the calculus of variations to prove things like the Neyman–Pearson lemma). A big gap in my knowledge is non-linear functional analysis beyond the vary basics of the calculus of variations so that might be a good place to start.