satisfies in particular Thus δ is a bounded linear functional on the Sobolev space H1. Equivalently δ is an element of the continuous dual space H−1 of H1. More generally, in n dimensions, one has provided . Spaces of holomorphic functions In complex analysis, the delta function enters via Cauchy's integral formula, which asserts that if D is a domain in the complex plane with smooth boundary, then for all holomorphic functions f in D that are continuous on the closure of D. As a result, the delta function δz is represented in this class of holomorphic functions by the Cauchy integral: Moreover, let H2(∂D) be the Hardy space consisting of the closure in L2(∂D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H2(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral formula continues to hold. In particular for , the delta function δz is a continuous linear functional on H2(∂D). This is a special case of the situation in several complex variables in which, for smooth domains D, the Szegő kernel plays the role of the Cauchy integral. Resolutions of the identity Given a complete orthonormal basis set of functions {φn} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as The coefficients {αn} are found as which may be represented by the notation: a form of the bra–ket notation of Dirac. Adopting this notation, the expansion of f takes the dyadic form: Letting I denote the identity operator on the Hilbert space, the expression is called a resolution of the identity. When the Hilbert space is the space L2(D) of square-integrable functions on a domain D, the quantity: is an integral operator, and the expression for f can be rewritten The right-hand side converges to f in the L2 sense. It need not hold in a pointwise sense, even when f is a continuous function. Nevertheless, it is common to abuse notation and write resulting in the representation of the delta function: With a suitable rigged Hilbert space where contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φn. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the distribution sense.