A Hilbert space is a complete inner-product space. Examples: $\ell^2$ (square-summable sequences), $L^2(\mathbb R)$ (square-integrable functions), and the QM state space.

Key structure: an inner product $\langle x, y\rangle$ induces a norm $\|x\| = \sqrt{\langle x, x\rangle}$ and satisfies Cauchy–Schwarz $|\langle x, y\rangle| \leq \|x\| \|y\|$. Complete in this norm. Orthonormal bases $\{e_n\}$ allow series expansions $x = \sum_n \langle e_n, x\rangle\, e_n$ (convergent in norm) with Parseval: $\|x\|^2 = \sum_n |\langle e_n, x\rangle|^2$.

A linear operator $T: H \to H$ is bounded if $\|T\| = \sup_{\|x\|=1} \|Tx\| < \infty$. For self-adjoint $T = T^*$, the spectral theorem generalizes diagonalization: $T = \int \lambda \, dE_\lambda$ for a projection-valued measure $E$. Compact self-adjoint operators have a discrete spectrum with eigenvectors forming an ONB — exactly what makes QM with the harmonic oscillator work.

Interactive: Fourier coefficients of $f$

Project a function onto the first $N$ Fourier basis vectors $\sin(n\pi x)$; reconstruct the partial sum.