Distributions and rigged Hilbert spaces

In the usual complex-linear distribution convention, a distribution T assigns a number T(φ) to each smooth compactly supported test function φ, linearly and continuously in the test-function topology. The delta at a acts by evaluation: δ_a(φ)=φ(a). It is not a function that happens to have infinite height and zero width. Differentiation is defined by transferring a derivative to the test function with a minus sign. This definition remains valid even when the original object has no pointwise derivative.

The topology is indispensable. Point evaluation is not a bounded functional on L2, because narrow unit-norm smooth bumps can have arbitrarily large values at a fixed point. On the test-function space it is continuous because the topology controls derivatives and values on compact sets. Thus a functional can be continuous on one space and not continuous on another. Calling something a dual vector without stating the test space and its topology leaves its continuity claim incomplete.

Take Φ to be the Schwartz space on the real line, equipped with seminorms controlling every derivative multiplied by every polynomial weight. It embeds densely and continuously in H=L2(R), but carries a stronger topology. For a rigged Hilbert triple compatible with our inner product, use Φ^×, the continuous conjugate-linear dual. A Hilbert vector f embeds as the functional F_f(φ)=inner-product(φ,f), which is conjugate linear in φ and linear in f. This produces linear embeddings Φ⊂H⊂Φ^×.

Mathematical distribution texts more often use the continuous linear dual Φ′. These conventions are related by conjugating the test function: F(φ)=T(conjugate φ). For example, the delta in the antidual acts as conjugate φ(a), rather than φ(a). Either convention is workable, but mixing them changes conjugations in generalized eigenvector formulas. The inclusions do not mean all dual elements are Hilbert vectors; they deliberately add generalized objects such as deltas and nondecaying oscillations.

For an operator A that maps Φ continuously into itself, a dual action can be defined by F↦[φ↦F(Aφ)]. For a symmetric differential operator this extends its Hilbert action in the antidual convention. Momentum P=−i d/dx preserves Schwartz functions. The functional associated formally with exp(ikx) obeys the generalized eigenvalue equation with eigenvalue k, verified by integration by parts against a rapidly decaying test function. The exponential is not in L2(R), so this is not a normalizable eigenvector of the Hilbert-space momentum operator.

A wave packet can instead be square integrable and expanded through a continuous Fourier transform with a stated normalization. Its spectral probability over an interval is an ordinary integral, rather than the squared norm of an individual plane wave. Completeness statements involving generalized eigenvectors require hypotheses on the operator and the chosen test space; not every arbitrary dense subspace automatically supplies a useful rigging. The finite periodic sine-sum lab illustrates legitimate finite functions. It cannot create a delta distribution, normalize a plane wave on the full line, or prove a generalized spectral expansion.