Functional analysis and operator domains

A Hilbert space is a complete inner-product space. In L2, the inner product integrates one function's complex conjugate times the other, with functions identified almost everywhere. Orthogonality, projection, and least-squares approximation still make sense. Completeness ensures that norm-convergent approximations remain inside the space. An orthonormal basis allows reconstruction through square-summable coefficients, but the resulting convergence is initially in norm; it does not automatically give pointwise values at every coordinate.

A bounded linear operator satisfies a uniform inequality relating output norm to input norm. Boundedness is equivalent to continuity for linear operators between normed spaces. It means one constant controls all inputs, not that the output itself is bounded independently of input size. On L2, multiplication by a bounded measurable function is bounded. An ideal point-evaluation functional is not generally well-defined there because changing one point does not change the L2 element.

On the periodic interval [0,2π], normalized waves e^(inx) have identical L2 norm, but their derivatives have norm proportional to |n|. No finite constant can bound all derivative outputs by the original L2 inputs. Differentiation is therefore unbounded in that norm. Moreover, an arbitrary L2 function need not possess an L2 weak derivative, so the derivative operator is not defined on the entire Hilbert space. A natural domain uses the Sobolev space H1 with the required boundary behavior.

An unbounded operator can still be well-behaved when densely defined and closed. Closedness means that if inputs and their operator outputs both converge in Hilbert norm, the limits remain a valid input-output pair. The graph norm combines the input norm and output norm to control both. These ideas replace the false expectation that every useful infinite-dimensional map behaves like a finite matrix. They also explain why truncating to finitely many frequencies hides unboundedness: each finite cutoff has a finite maximum frequency.

The formal momentum expression −i d/dx is not a complete operator specification. Integration by parts produces boundary terms, and the chosen domain determines whether those terms vanish and whether the adjoint has the same domain. Symmetry means the inner-product identity holds on the operator's domain. Self-adjointness additionally requires equality with the adjoint including its domain. For unbounded operators, symmetry alone does not guarantee the spectral and unitary-evolution conclusions used in quantum mechanics.

On a finite interval, periodic boundary conditions yield one self-adjoint momentum realization. More generally a phase-twisted periodic condition produces shifted momentum eigenvalues. Setting the wavefunction to zero at both endpoints makes the first-derivative expression symmetric on an appropriate dense domain but not self-adjoint. A self-adjoint Hamiltonian generates a strongly continuous unitary group; states outside its domain can still evolve unitarily even when a strong time derivative at zero is unavailable. Domain statements distinguish these claims precisely.