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True, but you introduce them first as dual vectors of the nuclear space in the rigged-Hilbert space formulation. I'd have to look up the precise mathematical construction of the corresponding kets and how to justify equations like ##\langle x|x' \rangle=\delta(x-x')##, where ##\delta## is the Dirac-##\delta## distribution. I guess you find it easily in the standard literature like Galindo&Pascual or de la Madrid.