unbounded operator

The most important operators in physics — differentiation, the momentum and energy operators of quantum mechanics — are not bounded, and cannot be made bounded. Differentiating a function can amplify it without limit: high-frequency wiggles get steeper and steeper under d/dx with no ceiling on the ratio ||T f|| / ||f||. An unbounded operator is one with no finite operator norm. The price you pay is that it cannot be defined on the whole space; you must track its domain.

Precisely: an unbounded operator T on a Hilbert space H is a linear map defined on a domain D(T), a dense linear subspace of H, with no constant C bounding ||T x|| <= C ||x||. Density of the domain is essential — it is what lets the adjoint be defined at all. Such operators are usually closed (their graph is closed) rather than continuous, which is the right substitute for boundedness in this setting.

Why the domain is not a technicality: two operators given by the same formula but different domains are genuinely different operators with different spectra and different adjoints. Whether d/dx is self-adjoint depends entirely on the boundary conditions baked into its domain. For unbounded operators, self-adjoint (T^* = T including equality of domains) is strictly stronger than symmetric (<T x, y> = <x, T y> on the domain), and only the self-adjoint ones generate the unitary time-evolution of quantum mechanics.

Why it matters: Stone's theorem and the spectral theorem extend to unbounded self-adjoint operators, which is exactly what quantum mechanics needs — position, momentum, and Hamiltonian operators are unbounded and self-adjoint, their real spectra being the possible measured values. Handling them rigorously is the reason domains, closedness, and self-adjointness are taken so seriously.