Foundations of Signal Processing
by Martin Vetterli, Jelena Kovačević, Vivek K. Goyal · Cambridge University Press
Cambridge University Press textbook by Vetterli, Kovačević, and Goyal, free as a PDF on the authors' site. Chapter 2, 'From Euclid to Hilbert', treats vector and Hilbert spaces, projections, bases, and frames, which later chapters use for Fourier analysis, sampling, and approximation. Covers the signal-processing side of the topic note, teaching Hilbert space geometry (projections, orthonormal bases, frames) as the working language of signals, with many worked examples.
More resources on Hilbert Spaces
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Hilbert Space
Advanced reference with category theory perspective, linked in r/math discussions.
Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators (2nd ed.)
Gerald Teschl's AMS graduate text, free as a PDF on his University of Vienna page. It develops Hilbert spaces, self-adjoint and unbounded operators, the spectral theorem, and Stone's theorem, then applies them to Schrödinger operators, the hydrogen atom, and scattering theory. Covers the quantum-mechanics side of the topic note with full proofs of the unbounded spectral theorem, where Kreyszig and Young stop.
Introductory Functional Analysis with Applications
Erwin Kreyszig's functional analysis textbook, written to need only linear algebra and basic analysis rather than measure theory. It covers inner product and Hilbert spaces, orthonormal sets, adjoint and self-adjoint operators, spectral theory, and applications of unbounded operators to quantum mechanics.
An Introduction to Hilbert Space
Nicholas Young's undergraduate text develops inner product spaces, orthogonal expansions, and Fourier series, then turns to linear operators, Sturm-Liouville systems, and Hardy spaces. Readers finish able to apply Hilbert space methods to differential equations and approximation problems.