Infinite-dimensional bicomplex Hilbert spaces

Abstract

‎This paper begins the study of infinite-dimensional‎ ‎modules defined on bicomplex numbers‎. ‎It generalizes‎ ‎a number of results obtained with finite-dimensional‎ ‎bicomplex modules‎. ‎The central concept introduced‎ ‎is the one of a bicomplex Hilbert space‎. ‎Properties‎ ‎of such spaces are obtained through properties of‎ ‎several of their subsets which have the structure of‎ ‎genuine Hilbert spaces‎. ‎In particular‎, ‎we derive the Riesz‎ ‎representation theorem for bicomplex continuous linear‎ ‎functionals and a general version of the bicomplex Schwarz‎ ‎inequality‎. ‎Applications to concepts relevant to quantum‎ ‎mechanics‎, ‎specifically the bicomplex analogue of the quantum‎ ‎harmonic oscillator‎, ‎are pointed out‎.